Find the Local Maxima and Minima f(x)=-2x^3-14x^2+2x+84

$f (x) = - 2 x^{3} - 14 x^{2} + 2 x + 84$

Step 1

Find the first derivative of the function.

By the Sum Rule, the derivative of $- 2 x^{3} - 14 x^{2} + 2 x + 84$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 14 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [84]$ .

$\frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 14 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [84]$

Evaluate $\frac{d}{d x} [- 2 x^{3}]$ .

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

$- 2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 14 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [84]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$- 2 (3 x^{2}) + \frac{d}{d x} [- 14 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [84]$

Multiply $3$ by $- 2$ .

$- 6 x^{2} + \frac{d}{d x} [- 14 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [84]$

Evaluate $\frac{d}{d x} [- 14 x^{2}]$ .

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x^{2}$ with respect to $x$ is $- 14 \frac{d}{d x} [x^{2}]$ .

$- 6 x^{2} - 14 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [84]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 6 x^{2} - 14 (2 x) + \frac{d}{d x} [2 x] + \frac{d}{d x} [84]$

Multiply $2$ by $- 14$ .

$- 6 x^{2} - 28 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [84]$

Evaluate $\frac{d}{d x} [2 x]$ .

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- 6 x^{2} - 28 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [84]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 6 x^{2} - 28 x + 2 \cdot 1 + \frac{d}{d x} [84]$

Multiply $2$ by $1$ .

$- 6 x^{2} - 28 x + 2 + \frac{d}{d x} [84]$

Differentiate using the Constant Rule.

Since $84$ is constant with respect to $x$ , the derivative of $84$ with respect to $x$ is $0$ .

$- 6 x^{2} - 28 x + 2 + 0$

Add $- 6 x^{2} - 28 x + 2$ and $0$ .

$- 6 x^{2} - 28 x + 2$

Step 2

Find the second derivative of the function.

By the Sum Rule, the derivative of $- 6 x^{2} - 28 x + 2$ with respect to $x$ is $\frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 28 x] + \frac{d}{d x} [2]$ .

$f'' (x) = \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (- 28 x) + \frac{d}{d x} (2)$

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - 6 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 28 x) + \frac{d}{d x} (2)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = - 6 (2 x) + \frac{d}{d x} (- 28 x) + \frac{d}{d x} (2)$

Multiply $2$ by $- 6$ .

$f'' (x) = - 12 x + \frac{d}{d x} (- 28 x) + \frac{d}{d x} (2)$

Evaluate $\frac{d}{d x} [- 28 x]$ .

Since $- 28$ is constant with respect to $x$ , the derivative of $- 28 x$ with respect to $x$ is $- 28 \frac{d}{d x} [x]$ .

$f'' (x) = - 12 x - 28 \frac{d}{d x} x + \frac{d}{d x} (2)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - 12 x - 28 \cdot 1 + \frac{d}{d x} (2)$

Multiply $- 28$ by $1$ .

$f'' (x) = - 12 x - 28 + \frac{d}{d x} (2)$

Differentiate using the Constant Rule.

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f'' (x) = - 12 x - 28 + 0$

Add $- 12 x - 28$ and $0$ .

$f'' (x) = - 12 x - 28$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- 6 x^{2} - 28 x + 2 = 0$

Step 4

Find the first derivative.

By the Sum Rule, the derivative of $- 2 x^{3} - 14 x^{2} + 2 x + 84$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 14 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [84]$ .

$f' (x) = \frac{d}{d x} (- 2 x^{3}) + \frac{d}{d x} (- 14 x^{2}) + \frac{d}{d x} (2 x) + \frac{d}{d x} (84)$

Evaluate $\frac{d}{d x} [- 2 x^{3}]$ .

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

$f' (x) = - 2 \frac{d}{d x} x^{3} + \frac{d}{d x} (- 14 x^{2}) + \frac{d}{d x} (2 x) + \frac{d}{d x} (84)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = - 2 (3 x^{2}) + \frac{d}{d x} (- 14 x^{2}) + \frac{d}{d x} (2 x) + \frac{d}{d x} (84)$

Multiply $3$ by $- 2$ .

$f' (x) = - 6 x^{2} + \frac{d}{d x} (- 14 x^{2}) + \frac{d}{d x} (2 x) + \frac{d}{d x} (84)$

Evaluate $\frac{d}{d x} [- 14 x^{2}]$ .

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x^{2}$ with respect to $x$ is $- 14 \frac{d}{d x} [x^{2}]$ .

$f' (x) = - 6 x^{2} - 14 \frac{d}{d x} x^{2} + \frac{d}{d x} (2 x) + \frac{d}{d x} (84)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = - 6 x^{2} - 14 (2 x) + \frac{d}{d x} (2 x) + \frac{d}{d x} (84)$

Multiply $2$ by $- 14$ .

$f' (x) = - 6 x^{2} - 28 x + \frac{d}{d x} (2 x) + \frac{d}{d x} (84)$

Evaluate $\frac{d}{d x} [2 x]$ .

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$f' (x) = - 6 x^{2} - 28 x + 2 \frac{d}{d x} (x) + \frac{d}{d x} (84)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = - 6 x^{2} - 28 x + 2 \cdot 1 + \frac{d}{d x} (84)$

Multiply $2$ by $1$ .

$f' (x) = - 6 x^{2} - 28 x + 2 + \frac{d}{d x} (84)$

Differentiate using the Constant Rule.

Since $84$ is constant with respect to $x$ , the derivative of $84$ with respect to $x$ is $0$ .

$f' (x) = - 6 x^{2} - 28 x + 2 + 0$

Add $- 6 x^{2} - 28 x + 2$ and $0$ .

$f' (x) = - 6 x^{2} - 28 x + 2$

The first derivative of $f (x)$ with respect to $x$ is $- 6 x^{2} - 28 x + 2$ .

$- 6 x^{2} - 28 x + 2$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 6 x^{2} - 28 x + 2 = 0$ .

Set the first derivative equal to $0$ .

$- 6 x^{2} - 28 x + 2 = 0$

Factor $- 2$ out of $- 6 x^{2} - 28 x + 2$ .

Factor $- 2$ out of $- 6 x^{2}$ .

$- 2 (3 x^{2}) - 28 x + 2 = 0$

Factor $- 2$ out of $- 28 x$ .

$- 2 (3 x^{2}) - 2 (14 x) + 2 = 0$

Factor $- 2$ out of $2$ .

$- 2 (3 x^{2}) - 2 (14 x) - 2 \cdot - 1 = 0$

Factor $- 2$ out of $- 2 (3 x^{2}) - 2 (14 x)$ .

$- 2 (3 x^{2} + 14 x) - 2 \cdot - 1 = 0$

Factor $- 2$ out of $- 2 (3 x^{2} + 14 x) - 2 (- 1)$ .

$- 2 (3 x^{2} + 14 x - 1) = 0$

Divide each term in $- 2 (3 x^{2} + 14 x - 1) = 0$ by $- 2$ and simplify.

Divide each term in $- 2 (3 x^{2} + 14 x - 1) = 0$ by $- 2$ .

$\frac{- 2 (3 x^{2} + 14 x - 1)}{- 2} = \frac{0}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 (3 x^{2} + 14 x - 1)}{- 2} = \frac{0}{- 2}$

Divide $3 x^{2} + 14 x - 1$ by $1$ .

$3 x^{2} + 14 x - 1 = \frac{0}{- 2}$

Simplify the right side.

Divide $0$ by $- 2$ .

$3 x^{2} + 14 x - 1 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 3$ , $b = 14$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (3 \cdot - 1)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 1$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 14 \pm \sqrt{196 - 12 \cdot - 1}}{2 \cdot 3}$

Multiply $- 12$ by $- 1$ .

$x = \frac{- 14 \pm \sqrt{196 + 12}}{2 \cdot 3}$

Add $196$ and $12$ .

$x = \frac{- 14 \pm \sqrt{208}}{2 \cdot 3}$

Rewrite $208$ as $4^{2} \cdot 13$ .

Factor $16$ out of $208$ .

$x = \frac{- 14 \pm \sqrt{16 (13)}}{2 \cdot 3}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 14 \pm \sqrt{4^{2} \cdot 13}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{- 14 \pm 4 \sqrt{13}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 14 \pm 4 \sqrt{13}}{6}$

Simplify $\frac{- 14 \pm 4 \sqrt{13}}{6}$ .

$x = \frac{- 7 \pm 2 \sqrt{13}}{3}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 1$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 14 \pm \sqrt{196 - 12 \cdot - 1}}{2 \cdot 3}$

Multiply $- 12$ by $- 1$ .

$x = \frac{- 14 \pm \sqrt{196 + 12}}{2 \cdot 3}$

Add $196$ and $12$ .

$x = \frac{- 14 \pm \sqrt{208}}{2 \cdot 3}$

Rewrite $208$ as $4^{2} \cdot 13$ .

Factor $16$ out of $208$ .

$x = \frac{- 14 \pm \sqrt{16 (13)}}{2 \cdot 3}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 14 \pm \sqrt{4^{2} \cdot 13}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{- 14 \pm 4 \sqrt{13}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 14 \pm 4 \sqrt{13}}{6}$

Simplify $\frac{- 14 \pm 4 \sqrt{13}}{6}$ .

$x = \frac{- 7 \pm 2 \sqrt{13}}{3}$

Change the $\pm$ to $+$ .

$x = \frac{- 7 + 2 \sqrt{13}}{3}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + 2 \sqrt{13}}{3}$

Factor $- 1$ out of $2 \sqrt{13}$ .

$x = \frac{- 1 \cdot 7 - (- 2 \sqrt{13})}{3}$

Factor $- 1$ out of $- 1 (7) - (- 2 \sqrt{13})$ .

$x = \frac{- 1 (7 - 2 \sqrt{13})}{3}$

Move the negative in front of the fraction.

$x = - \frac{7 - 2 \sqrt{13}}{3}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 1$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 14 \pm \sqrt{196 - 12 \cdot - 1}}{2 \cdot 3}$

Multiply $- 12$ by $- 1$ .

$x = \frac{- 14 \pm \sqrt{196 + 12}}{2 \cdot 3}$

Add $196$ and $12$ .

$x = \frac{- 14 \pm \sqrt{208}}{2 \cdot 3}$

Rewrite $208$ as $4^{2} \cdot 13$ .

Factor $16$ out of $208$ .

$x = \frac{- 14 \pm \sqrt{16 (13)}}{2 \cdot 3}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 14 \pm \sqrt{4^{2} \cdot 13}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{- 14 \pm 4 \sqrt{13}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 14 \pm 4 \sqrt{13}}{6}$

Simplify $\frac{- 14 \pm 4 \sqrt{13}}{6}$ .

$x = \frac{- 7 \pm 2 \sqrt{13}}{3}$

Change the $\pm$ to $-$ .

$x = \frac{- 7 - 2 \sqrt{13}}{3}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 - 2 \sqrt{13}}{3}$

Factor $- 1$ out of $- 2 \sqrt{13}$ .

$x = \frac{- 1 \cdot 7 - (2 \sqrt{13})}{3}$

Factor $- 1$ out of $- 1 (7) - (2 \sqrt{13})$ .

$x = \frac{- 1 (7 + 2 \sqrt{13})}{3}$

Move the negative in front of the fraction.

$x = - \frac{7 + 2 \sqrt{13}}{3}$

The final answer is the combination of both solutions.

$x = - \frac{7 - 2 \sqrt{13}}{3}, - \frac{7 + 2 \sqrt{13}}{3}$

Step 6

Find the values where the derivative is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = - \frac{7 - 2 \sqrt{13}}{3}, - \frac{7 + 2 \sqrt{13}}{3}$

Step 8

Evaluate the second derivative at $x = - \frac{7 - 2 \sqrt{13}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 12 (- \frac{7 - 2 \sqrt{13}}{3}) - 28$

Step 9

Evaluate the second derivative.

Simplify each term.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{7 - 2 \sqrt{13}}{3}$ into the numerator.

$- 12 \frac{- (7 - 2 \sqrt{13})}{3} - 28$

Factor $3$ out of $- 12$ .

$3 (- 4) \frac{- (7 - 2 \sqrt{13})}{3} - 28$

Cancel the common factor.

$3 \cdot - 4 \frac{- (7 - 2 \sqrt{13})}{3} - 28$

Rewrite the expression.

$- 4 (- (7 - 2 \sqrt{13})) - 28$

Multiply $- 1$ by $- 4$ .

$4 (7 - 2 \sqrt{13}) - 28$

Apply the distributive property.

$4 \cdot 7 + 4 (- 2 \sqrt{13}) - 28$

Multiply $4$ by $7$ .

$28 + 4 (- 2 \sqrt{13}) - 28$

Multiply $- 2$ by $4$ .

$28 - 8 \sqrt{13} - 28$

Simplify by subtracting numbers.

Subtract $28$ from $28$ .

$0 - 8 \sqrt{13}$

Subtract $8 \sqrt{13}$ from $0$ .

$- 8 \sqrt{13}$

Step 10

$x = - \frac{7 - 2 \sqrt{13}}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{7 - 2 \sqrt{13}}{3}$ is a local maximum

Step 11

Find the y-value when $x = - \frac{7 - 2 \sqrt{13}}{3}$ .

Replace the variable $x$ with $- \frac{7 - 2 \sqrt{13}}{3}$ in the expression.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 {(- \frac{7 - 2 \sqrt{13}}{3})}^{3} - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Simplify the result.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{7 - 2 \sqrt{13}}{3}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 ({(- 1)}^{3} {(\frac{7 - 2 \sqrt{13}}{3})}^{3}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Apply the product rule to $\frac{7 - 2 \sqrt{13}}{3}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 ({(- 1)}^{3} (\frac{{(7 - 2 \sqrt{13})}^{3}}{3^{3}})) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Raise $- 1$ to the power of $3$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{{(7 - 2 \sqrt{13})}^{3}}{3^{3}}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Raise $3$ to the power of $3$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{{(7 - 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Use the Binomial Theorem.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{7^{3} + 3 \cdot (7^{2} (- 2 \sqrt{13})) + 3 \cdot (7 {(- 2 \sqrt{13})}^{2}) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Simplify each term.

Raise $7$ to the power of $3$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 3 \cdot (7^{2} (- 2 \sqrt{13})) + 3 \cdot (7 {(- 2 \sqrt{13})}^{2}) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Raise $7$ to the power of $2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 3 \cdot (49 (- 2 \sqrt{13})) + 3 \cdot (7 {(- 2 \sqrt{13})}^{2}) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $3$ by $49$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 147 (- 2 \sqrt{13}) + 3 \cdot (7 {(- 2 \sqrt{13})}^{2}) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $- 2$ by $147$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 3 \cdot (7 {(- 2 \sqrt{13})}^{2}) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $3$ by $7$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 21 {(- 2 \sqrt{13})}^{2} + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Apply the product rule to $- 2 \sqrt{13}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 21 ({(- 2)}^{2} {\sqrt{13}}^{2}) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Raise $- 2$ to the power of $2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 21 (4 {\sqrt{13}}^{2}) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 21 (4 {(13^{\frac{1}{2}})}^{2}) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 21 (4 \cdot 13^{\frac{1}{2} \cdot 2}) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 21 (4 \cdot 13^{\frac{2}{2}}) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Cancel the common factor of $2$ .

Cancel the common factor.

Rewrite the expression.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 21 (4 \cdot 13) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Evaluate the exponent.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 21 (4 \cdot 13) + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $21 (4 \cdot 13)$ .

Multiply $4$ by $13$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 21 \cdot 52 + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $21$ by $52$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 1092 + {(- 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Apply the product rule to $- 2 \sqrt{13}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 1092 + {(- 2)}^{3} {\sqrt{13}}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Raise $- 2$ to the power of $3$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 1092 - 8 {\sqrt{13}}^{3}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Rewrite ${\sqrt{13}}^{3}$ as $\sqrt{13^{3}}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 1092 - 8 \sqrt{13^{3}}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Raise $13$ to the power of $3$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 1092 - 8 \sqrt{2197}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Rewrite $2197$ as $13^{2} \cdot 13$ .

Factor $169$ out of $2197$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 1092 - 8 \sqrt{169 (13)}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Rewrite $169$ as $13^{2}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 1092 - 8 \sqrt{13^{2} \cdot 13}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Pull terms out from under the radical.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 1092 - 8 (13 \sqrt{13})}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $13$ by $- 8$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{343 - 294 \sqrt{13} + 1092 - 104 \sqrt{13}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Add $343$ and $1092$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{1435 - 294 \sqrt{13} - 104 \sqrt{13}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Subtract $104 \sqrt{13}$ from $- 294 \sqrt{13}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = - 2 (- \frac{1435 - 398 \sqrt{13}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $- 2 (- \frac{1435 - 398 \sqrt{13}}{27})$ .

Multiply $- 1$ by $- 2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = 2 (\frac{1435 - 398 \sqrt{13}}{27}) - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Combine $2$ and $\frac{1435 - 398 \sqrt{13}}{27}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 {(- \frac{7 - 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{7 - 2 \sqrt{13}}{3}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 ({(- 1)}^{2} {(\frac{7 - 2 \sqrt{13}}{3})}^{2}) + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Apply the product rule to $\frac{7 - 2 \sqrt{13}}{3}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 ({(- 1)}^{2} (\frac{{(7 - 2 \sqrt{13})}^{2}}{3^{2}})) + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Raise $- 1$ to the power of $2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 (1 (\frac{{(7 - 2 \sqrt{13})}^{2}}{3^{2}})) + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $\frac{{(7 - 2 \sqrt{13})}^{2}}{3^{2}}$ by $1$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{{(7 - 2 \sqrt{13})}^{2}}{3^{2}} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Raise $3$ to the power of $2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{{(7 - 2 \sqrt{13})}^{2}}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Rewrite ${(7 - 2 \sqrt{13})}^{2}$ as $(7 - 2 \sqrt{13}) (7 - 2 \sqrt{13})$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{(7 - 2 \sqrt{13}) (7 - 2 \sqrt{13})}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Expand $(7 - 2 \sqrt{13}) (7 - 2 \sqrt{13})$ using the FOIL Method.

Apply the distributive property.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{7 (7 - 2 \sqrt{13}) - 2 \sqrt{13} (7 - 2 \sqrt{13})}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Apply the distributive property.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{7 \cdot 7 + 7 (- 2 \sqrt{13}) - 2 \sqrt{13} (7 - 2 \sqrt{13})}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Apply the distributive property.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{7 \cdot 7 + 7 (- 2 \sqrt{13}) - 2 \sqrt{13} \cdot 7 - 2 \sqrt{13} (- 2 \sqrt{13})}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 + 7 (- 2 \sqrt{13}) - 2 \sqrt{13} \cdot 7 - 2 \sqrt{13} (- 2 \sqrt{13})}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $- 2$ by $7$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 2 \sqrt{13} \cdot 7 - 2 \sqrt{13} (- 2 \sqrt{13})}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $7$ by $- 2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} - 2 \sqrt{13} (- 2 \sqrt{13})}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $- 2 \sqrt{13} (- 2 \sqrt{13})$ .

Multiply $- 2$ by $- 2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 4 \sqrt{13} \sqrt{13}}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Raise $\sqrt{13}$ to the power of $1$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 4 (\sqrt{13} \sqrt{13})}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Raise $\sqrt{13}$ to the power of $1$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 4 (\sqrt{13} \sqrt{13})}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 4 {\sqrt{13}}^{1 + 1}}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Add $1$ and $1$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 4 {\sqrt{13}}^{2}}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 4 {(13^{\frac{1}{2}})}^{2}}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 4 \cdot 13^{\frac{1}{2} \cdot 2}}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 4 \cdot 13^{\frac{2}{2}}}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 4 \cdot 13^{\frac{2}{2}}}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Rewrite the expression.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 4 \cdot 13}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Evaluate the exponent.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 4 \cdot 13}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $4$ by $13$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{49 - 14 \sqrt{13} - 14 \sqrt{13} + 52}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Add $49$ and $52$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{101 - 14 \sqrt{13} - 14 \sqrt{13}}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Subtract $14 \sqrt{13}$ from $- 14 \sqrt{13}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - 14 \frac{101 - 28 \sqrt{13}}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Combine $- 14$ and $\frac{101 - 28 \sqrt{13}}{9}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} + \frac{- 14 (101 - 28 \sqrt{13})}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Move the negative in front of the fraction.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{(14) (101 - 28 \sqrt{13})}{9} + 2 (- \frac{7 - 2 \sqrt{13}}{3}) + 84$

Multiply $2 (- \frac{7 - 2 \sqrt{13}}{3})$ .

Multiply $- 1$ by $2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13})}{9} - 2 \frac{7 - 2 \sqrt{13}}{3} + 84$

Combine $- 2$ and $\frac{7 - 2 \sqrt{13}}{3}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13})}{9} + \frac{- 2 (7 - 2 \sqrt{13})}{3} + 84$

Move the negative in front of the fraction.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13})}{9} - \frac{2 (7 - 2 \sqrt{13})}{3} + 84$

Find the common denominator.

Multiply $\frac{14 (101 - 28 \sqrt{13})}{9}$ by $\frac{3}{3}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - (\frac{14 (101 - 28 \sqrt{13})}{9} \cdot \frac{3}{3}) - \frac{2 (7 - 2 \sqrt{13})}{3} + 84$

Multiply $\frac{14 (101 - 28 \sqrt{13})}{9}$ by $\frac{3}{3}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - \frac{2 (7 - 2 \sqrt{13})}{3} + 84$

Multiply $\frac{2 (7 - 2 \sqrt{13})}{3}$ by $\frac{9}{9}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - (\frac{2 (7 - 2 \sqrt{13})}{3} \cdot \frac{9}{9}) + 84$

Multiply $\frac{2 (7 - 2 \sqrt{13})}{3}$ by $\frac{9}{9}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - \frac{2 (7 - 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + 84$

Write $84$ as a fraction with denominator $1$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - \frac{2 (7 - 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + \frac{84}{1}$

Multiply $\frac{84}{1}$ by $\frac{27}{27}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - \frac{2 (7 - 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + \frac{84}{1} \cdot \frac{27}{27}$

Multiply $\frac{84}{1}$ by $\frac{27}{27}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - \frac{2 (7 - 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + \frac{84 \cdot 27}{27}$

Reorder the factors of $9 \cdot 3$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13}) \cdot 3}{3 \cdot 9} - \frac{2 (7 - 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + \frac{84 \cdot 27}{27}$

Multiply $3$ by $9$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13}) \cdot 3}{27} - \frac{2 (7 - 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + \frac{84 \cdot 27}{27}$

Multiply $3$ by $9$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13})}{27} - \frac{14 (101 - 28 \sqrt{13}) \cdot 3}{27} - \frac{2 (7 - 2 \sqrt{13}) \cdot 9}{27} + \frac{84 \cdot 27}{27}$

Combine the numerators over the common denominator.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 (1435 - 398 \sqrt{13}) - 14 (101 - 28 \sqrt{13}) \cdot 3 - 2 (7 - 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Simplify each term.

Apply the distributive property.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2 \cdot 1435 + 2 (- 398 \sqrt{13}) - 14 (101 - 28 \sqrt{13}) \cdot 3 - 2 (7 - 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $2$ by $1435$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 + 2 (- 398 \sqrt{13}) - 14 (101 - 28 \sqrt{13}) \cdot 3 - 2 (7 - 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $- 398$ by $2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} - 14 (101 - 28 \sqrt{13}) \cdot 3 - 2 (7 - 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Apply the distributive property.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} + (- 14 \cdot 101 - 14 (- 28 \sqrt{13})) \cdot 3 - 2 (7 - 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $- 14$ by $101$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} + (- 1414 - 14 (- 28 \sqrt{13})) \cdot 3 - 2 (7 - 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $- 28$ by $- 14$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} + (- 1414 + 392 \sqrt{13}) \cdot 3 - 2 (7 - 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Apply the distributive property.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} - 1414 \cdot 3 + 392 \sqrt{13} \cdot 3 - 2 (7 - 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $- 1414$ by $3$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} - 4242 + 392 \sqrt{13} \cdot 3 - 2 (7 - 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $3$ by $392$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} - 4242 + 1176 \sqrt{13} - 2 (7 - 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Apply the distributive property.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} - 4242 + 1176 \sqrt{13} + (- 2 \cdot 7 - 2 (- 2 \sqrt{13})) \cdot 9 + 84 \cdot 27}{27}$

Multiply $- 2$ by $7$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} - 4242 + 1176 \sqrt{13} + (- 14 - 2 (- 2 \sqrt{13})) \cdot 9 + 84 \cdot 27}{27}$

Multiply $- 2$ by $- 2$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} - 4242 + 1176 \sqrt{13} + (- 14 + 4 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Apply the distributive property.

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} - 4242 + 1176 \sqrt{13} - 14 \cdot 9 + 4 \sqrt{13} \cdot 9 + 84 \cdot 27}{27}$

Multiply $- 14$ by $9$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} - 4242 + 1176 \sqrt{13} - 126 + 4 \sqrt{13} \cdot 9 + 84 \cdot 27}{27}$

Multiply $9$ by $4$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} - 4242 + 1176 \sqrt{13} - 126 + 36 \sqrt{13} + 84 \cdot 27}{27}$

Multiply $84$ by $27$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{2870 - 796 \sqrt{13} - 4242 + 1176 \sqrt{13} - 126 + 36 \sqrt{13} + 2268}{27}$

Simplify by adding terms.

Subtract $4242$ from $2870$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{- 1372 - 796 \sqrt{13} + 1176 \sqrt{13} - 126 + 36 \sqrt{13} + 2268}{27}$

Simplify by adding and subtracting.

Subtract $126$ from $- 1372$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{- 1498 - 796 \sqrt{13} + 1176 \sqrt{13} + 36 \sqrt{13} + 2268}{27}$

Add $- 1498$ and $2268$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{770 - 796 \sqrt{13} + 1176 \sqrt{13} + 36 \sqrt{13}}{27}$

Add $- 796 \sqrt{13}$ and $1176 \sqrt{13}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{770 + 380 \sqrt{13} + 36 \sqrt{13}}{27}$

Add $380 \sqrt{13}$ and $36 \sqrt{13}$ .

$f (- \frac{7 - 2 \sqrt{13}}{3}) = \frac{770 + 416 \sqrt{13}}{27}$

The final answer is $\frac{770 + 416 \sqrt{13}}{27}$ .

$y = \frac{770 + 416 \sqrt{13}}{27}$

Step 12

Evaluate the second derivative at $x = - \frac{7 + 2 \sqrt{13}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 12 (- \frac{7 + 2 \sqrt{13}}{3}) - 28$

Step 13

Evaluate the second derivative.

Simplify each term.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{7 + 2 \sqrt{13}}{3}$ into the numerator.

$- 12 \frac{- (7 + 2 \sqrt{13})}{3} - 28$

Factor $3$ out of $- 12$ .

$3 (- 4) \frac{- (7 + 2 \sqrt{13})}{3} - 28$

Cancel the common factor.

$3 \cdot - 4 \frac{- (7 + 2 \sqrt{13})}{3} - 28$

Rewrite the expression.

$- 4 (- (7 + 2 \sqrt{13})) - 28$

Multiply $- 1$ by $- 4$ .

$4 (7 + 2 \sqrt{13}) - 28$

Apply the distributive property.

$4 \cdot 7 + 4 (2 \sqrt{13}) - 28$

Multiply $4$ by $7$ .

$28 + 4 (2 \sqrt{13}) - 28$

Multiply $2$ by $4$ .

$28 + 8 \sqrt{13} - 28$

Simplify by subtracting numbers.

Subtract $28$ from $28$ .

$0 + 8 \sqrt{13}$

Add $0$ and $8 \sqrt{13}$ .

$8 \sqrt{13}$

Step 14

$x = - \frac{7 + 2 \sqrt{13}}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{7 + 2 \sqrt{13}}{3}$ is a local minimum

Step 15

Find the y-value when $x = - \frac{7 + 2 \sqrt{13}}{3}$ .

Replace the variable $x$ with $- \frac{7 + 2 \sqrt{13}}{3}$ in the expression.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 {(- \frac{7 + 2 \sqrt{13}}{3})}^{3} - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Simplify the result.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{7 + 2 \sqrt{13}}{3}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 ({(- 1)}^{3} {(\frac{7 + 2 \sqrt{13}}{3})}^{3}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Apply the product rule to $\frac{7 + 2 \sqrt{13}}{3}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 ({(- 1)}^{3} (\frac{{(7 + 2 \sqrt{13})}^{3}}{3^{3}})) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Raise $- 1$ to the power of $3$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{{(7 + 2 \sqrt{13})}^{3}}{3^{3}}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Raise $3$ to the power of $3$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{{(7 + 2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Use the Binomial Theorem.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{7^{3} + 3 \cdot (7^{2} (2 \sqrt{13})) + 3 \cdot (7 {(2 \sqrt{13})}^{2}) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Simplify each term.

Raise $7$ to the power of $3$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 3 \cdot (7^{2} (2 \sqrt{13})) + 3 \cdot (7 {(2 \sqrt{13})}^{2}) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Raise $7$ to the power of $2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 3 \cdot (49 (2 \sqrt{13})) + 3 \cdot (7 {(2 \sqrt{13})}^{2}) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $3$ by $49$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 147 (2 \sqrt{13}) + 3 \cdot (7 {(2 \sqrt{13})}^{2}) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $2$ by $147$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 3 \cdot (7 {(2 \sqrt{13})}^{2}) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $3$ by $7$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 21 {(2 \sqrt{13})}^{2} + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Apply the product rule to $2 \sqrt{13}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 21 (2^{2} {\sqrt{13}}^{2}) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Raise $2$ to the power of $2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 21 (4 {\sqrt{13}}^{2}) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 21 (4 {(13^{\frac{1}{2}})}^{2}) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 21 (4 \cdot 13^{\frac{1}{2} \cdot 2}) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 21 (4 \cdot 13^{\frac{2}{2}}) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Cancel the common factor of $2$ .

Cancel the common factor.

Rewrite the expression.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 21 (4 \cdot 13) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Evaluate the exponent.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 21 (4 \cdot 13) + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $21 (4 \cdot 13)$ .

Multiply $4$ by $13$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 21 \cdot 52 + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $21$ by $52$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 1092 + {(2 \sqrt{13})}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Apply the product rule to $2 \sqrt{13}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 1092 + 2^{3} {\sqrt{13}}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Raise $2$ to the power of $3$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 1092 + 8 {\sqrt{13}}^{3}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Rewrite ${\sqrt{13}}^{3}$ as $\sqrt{13^{3}}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 1092 + 8 \sqrt{13^{3}}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Raise $13$ to the power of $3$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 1092 + 8 \sqrt{2197}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Rewrite $2197$ as $13^{2} \cdot 13$ .

Factor $169$ out of $2197$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 1092 + 8 \sqrt{169 (13)}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Rewrite $169$ as $13^{2}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 1092 + 8 \sqrt{13^{2} \cdot 13}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Pull terms out from under the radical.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 1092 + 8 (13 \sqrt{13})}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $13$ by $8$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{343 + 294 \sqrt{13} + 1092 + 104 \sqrt{13}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Add $343$ and $1092$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{1435 + 294 \sqrt{13} + 104 \sqrt{13}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Add $294 \sqrt{13}$ and $104 \sqrt{13}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = - 2 (- \frac{1435 + 398 \sqrt{13}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $- 2 (- \frac{1435 + 398 \sqrt{13}}{27})$ .

Multiply $- 1$ by $- 2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = 2 (\frac{1435 + 398 \sqrt{13}}{27}) - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Combine $2$ and $\frac{1435 + 398 \sqrt{13}}{27}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 {(- \frac{7 + 2 \sqrt{13}}{3})}^{2} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{7 + 2 \sqrt{13}}{3}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 ({(- 1)}^{2} {(\frac{7 + 2 \sqrt{13}}{3})}^{2}) + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Apply the product rule to $\frac{7 + 2 \sqrt{13}}{3}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 ({(- 1)}^{2} (\frac{{(7 + 2 \sqrt{13})}^{2}}{3^{2}})) + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Raise $- 1$ to the power of $2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 (1 (\frac{{(7 + 2 \sqrt{13})}^{2}}{3^{2}})) + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $\frac{{(7 + 2 \sqrt{13})}^{2}}{3^{2}}$ by $1$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{{(7 + 2 \sqrt{13})}^{2}}{3^{2}} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Raise $3$ to the power of $2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{{(7 + 2 \sqrt{13})}^{2}}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Rewrite ${(7 + 2 \sqrt{13})}^{2}$ as $(7 + 2 \sqrt{13}) (7 + 2 \sqrt{13})$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{(7 + 2 \sqrt{13}) (7 + 2 \sqrt{13})}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Expand $(7 + 2 \sqrt{13}) (7 + 2 \sqrt{13})$ using the FOIL Method.

Apply the distributive property.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{7 (7 + 2 \sqrt{13}) + 2 \sqrt{13} (7 + 2 \sqrt{13})}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Apply the distributive property.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{7 \cdot 7 + 7 (2 \sqrt{13}) + 2 \sqrt{13} (7 + 2 \sqrt{13})}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Apply the distributive property.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{7 \cdot 7 + 7 (2 \sqrt{13}) + 2 \sqrt{13} \cdot 7 + 2 \sqrt{13} (2 \sqrt{13})}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 7 (2 \sqrt{13}) + 2 \sqrt{13} \cdot 7 + 2 \sqrt{13} (2 \sqrt{13})}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $2$ by $7$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 2 \sqrt{13} \cdot 7 + 2 \sqrt{13} (2 \sqrt{13})}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $7$ by $2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 2 \sqrt{13} (2 \sqrt{13})}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $2 \sqrt{13} (2 \sqrt{13})$ .

Multiply $2$ by $2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 4 \sqrt{13} \sqrt{13}}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Raise $\sqrt{13}$ to the power of $1$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 4 (\sqrt{13} \sqrt{13})}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Raise $\sqrt{13}$ to the power of $1$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 4 (\sqrt{13} \sqrt{13})}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 4 {\sqrt{13}}^{1 + 1}}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Add $1$ and $1$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 4 {\sqrt{13}}^{2}}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 4 {(13^{\frac{1}{2}})}^{2}}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 4 \cdot 13^{\frac{1}{2} \cdot 2}}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 4 \cdot 13^{\frac{2}{2}}}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 4 \cdot 13^{\frac{2}{2}}}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Rewrite the expression.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 4 \cdot 13}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Evaluate the exponent.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 4 \cdot 13}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $4$ by $13$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{49 + 14 \sqrt{13} + 14 \sqrt{13} + 52}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Add $49$ and $52$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{101 + 14 \sqrt{13} + 14 \sqrt{13}}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Add $14 \sqrt{13}$ and $14 \sqrt{13}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - 14 \frac{101 + 28 \sqrt{13}}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Combine $- 14$ and $\frac{101 + 28 \sqrt{13}}{9}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} + \frac{- 14 (101 + 28 \sqrt{13})}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Move the negative in front of the fraction.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{(14) (101 + 28 \sqrt{13})}{9} + 2 (- \frac{7 + 2 \sqrt{13}}{3}) + 84$

Multiply $2 (- \frac{7 + 2 \sqrt{13}}{3})$ .

Multiply $- 1$ by $2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13})}{9} - 2 \frac{7 + 2 \sqrt{13}}{3} + 84$

Combine $- 2$ and $\frac{7 + 2 \sqrt{13}}{3}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13})}{9} + \frac{- 2 (7 + 2 \sqrt{13})}{3} + 84$

Move the negative in front of the fraction.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13})}{9} - \frac{2 (7 + 2 \sqrt{13})}{3} + 84$

Find the common denominator.

Multiply $\frac{14 (101 + 28 \sqrt{13})}{9}$ by $\frac{3}{3}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - (\frac{14 (101 + 28 \sqrt{13})}{9} \cdot \frac{3}{3}) - \frac{2 (7 + 2 \sqrt{13})}{3} + 84$

Multiply $\frac{14 (101 + 28 \sqrt{13})}{9}$ by $\frac{3}{3}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - \frac{2 (7 + 2 \sqrt{13})}{3} + 84$

Multiply $\frac{2 (7 + 2 \sqrt{13})}{3}$ by $\frac{9}{9}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - (\frac{2 (7 + 2 \sqrt{13})}{3} \cdot \frac{9}{9}) + 84$

Multiply $\frac{2 (7 + 2 \sqrt{13})}{3}$ by $\frac{9}{9}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - \frac{2 (7 + 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + 84$

Write $84$ as a fraction with denominator $1$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - \frac{2 (7 + 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + \frac{84}{1}$

Multiply $\frac{84}{1}$ by $\frac{27}{27}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - \frac{2 (7 + 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + \frac{84}{1} \cdot \frac{27}{27}$

Multiply $\frac{84}{1}$ by $\frac{27}{27}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13}) \cdot 3}{9 \cdot 3} - \frac{2 (7 + 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + \frac{84 \cdot 27}{27}$

Reorder the factors of $9 \cdot 3$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13}) \cdot 3}{3 \cdot 9} - \frac{2 (7 + 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + \frac{84 \cdot 27}{27}$

Multiply $3$ by $9$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13}) \cdot 3}{27} - \frac{2 (7 + 2 \sqrt{13}) \cdot 9}{3 \cdot 9} + \frac{84 \cdot 27}{27}$

Multiply $3$ by $9$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13})}{27} - \frac{14 (101 + 28 \sqrt{13}) \cdot 3}{27} - \frac{2 (7 + 2 \sqrt{13}) \cdot 9}{27} + \frac{84 \cdot 27}{27}$

Combine the numerators over the common denominator.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 (1435 + 398 \sqrt{13}) - 14 (101 + 28 \sqrt{13}) \cdot 3 - 2 (7 + 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Simplify each term.

Apply the distributive property.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2 \cdot 1435 + 2 (398 \sqrt{13}) - 14 (101 + 28 \sqrt{13}) \cdot 3 - 2 (7 + 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $2$ by $1435$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 2 (398 \sqrt{13}) - 14 (101 + 28 \sqrt{13}) \cdot 3 - 2 (7 + 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $398$ by $2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} - 14 (101 + 28 \sqrt{13}) \cdot 3 - 2 (7 + 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Apply the distributive property.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} + (- 14 \cdot 101 - 14 (28 \sqrt{13})) \cdot 3 - 2 (7 + 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $- 14$ by $101$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} + (- 1414 - 14 (28 \sqrt{13})) \cdot 3 - 2 (7 + 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $28$ by $- 14$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} + (- 1414 - 392 \sqrt{13}) \cdot 3 - 2 (7 + 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Apply the distributive property.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} - 1414 \cdot 3 - 392 \sqrt{13} \cdot 3 - 2 (7 + 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $- 1414$ by $3$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} - 4242 - 392 \sqrt{13} \cdot 3 - 2 (7 + 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Multiply $3$ by $- 392$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} - 4242 - 1176 \sqrt{13} - 2 (7 + 2 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Apply the distributive property.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} - 4242 - 1176 \sqrt{13} + (- 2 \cdot 7 - 2 (2 \sqrt{13})) \cdot 9 + 84 \cdot 27}{27}$

Multiply $- 2$ by $7$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} - 4242 - 1176 \sqrt{13} + (- 14 - 2 (2 \sqrt{13})) \cdot 9 + 84 \cdot 27}{27}$

Multiply $2$ by $- 2$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} - 4242 - 1176 \sqrt{13} + (- 14 - 4 \sqrt{13}) \cdot 9 + 84 \cdot 27}{27}$

Apply the distributive property.

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} - 4242 - 1176 \sqrt{13} - 14 \cdot 9 - 4 \sqrt{13} \cdot 9 + 84 \cdot 27}{27}$

Multiply $- 14$ by $9$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} - 4242 - 1176 \sqrt{13} - 126 - 4 \sqrt{13} \cdot 9 + 84 \cdot 27}{27}$

Multiply $9$ by $- 4$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} - 4242 - 1176 \sqrt{13} - 126 - 36 \sqrt{13} + 84 \cdot 27}{27}$

Multiply $84$ by $27$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{2870 + 796 \sqrt{13} - 4242 - 1176 \sqrt{13} - 126 - 36 \sqrt{13} + 2268}{27}$

Simplify by adding terms.

Subtract $4242$ from $2870$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{- 1372 + 796 \sqrt{13} - 1176 \sqrt{13} - 126 - 36 \sqrt{13} + 2268}{27}$

Simplify by adding and subtracting.

Subtract $126$ from $- 1372$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{- 1498 + 796 \sqrt{13} - 1176 \sqrt{13} - 36 \sqrt{13} + 2268}{27}$

Add $- 1498$ and $2268$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{770 + 796 \sqrt{13} - 1176 \sqrt{13} - 36 \sqrt{13}}{27}$

Subtract $1176 \sqrt{13}$ from $796 \sqrt{13}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{770 - 380 \sqrt{13} - 36 \sqrt{13}}{27}$

Subtract $36 \sqrt{13}$ from $- 380 \sqrt{13}$ .

$f (- \frac{7 + 2 \sqrt{13}}{3}) = \frac{770 - 416 \sqrt{13}}{27}$

The final answer is $\frac{770 - 416 \sqrt{13}}{27}$ .

$y = \frac{770 - 416 \sqrt{13}}{27}$

Step 16

These are the local extrema for $f (x) = - 2 x^{3} - 14 x^{2} + 2 x + 84$ .

$(- \frac{7 - 2 \sqrt{13}}{3}, \frac{770 + 416 \sqrt{13}}{27})$ is a local maxima

$(- \frac{7 + 2 \sqrt{13}}{3}, \frac{770 - 416 \sqrt{13}}{27})$ is a local minima

Step 17

Find the Local Maxima and Minima f(x)=-2x^3-14x^2+2x+84

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