Find the Local Maxima and Minima x^3+2x^2-5x

$x^{3} + 2 x^{2} - 5 x$

Step 1

Write $x^{3} + 2 x^{2} - 5 x$ as a function.

$f (x) = x^{3} + 2 x^{2} - 5 x$

Step 2

Find the first derivative of the function.

Differentiate.

By the Sum Rule, the derivative of $x^{3} + 2 x^{2} - 5 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x]$

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

$3 x^{2} + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x]$

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

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

$3 x^{2} + 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x]$

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

$3 x^{2} + 2 (2 x) + \frac{d}{d x} [- 5 x]$

Multiply $2$ by $2$ .

$3 x^{2} + 4 x + \frac{d}{d x} [- 5 x]$

Evaluate $\frac{d}{d x} [- 5 x]$ .

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$3 x^{2} + 4 x - 5 \frac{d}{d x} [x]$

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

$3 x^{2} + 4 x - 5 \cdot 1$

Multiply $- 5$ by $1$ .

$3 x^{2} + 4 x - 5$

Step 3

Find the second derivative of the function.

By the Sum Rule, the derivative of $3 x^{2} + 4 x - 5$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 5]$ .

$f'' (x) = \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (4 x) + \frac{d}{d x} (- 5)$

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

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

$f'' (x) = 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (4 x) + \frac{d}{d x} (- 5)$

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

$f'' (x) = 3 (2 x) + \frac{d}{d x} (4 x) + \frac{d}{d x} (- 5)$

Multiply $2$ by $3$ .

$f'' (x) = 6 x + \frac{d}{d x} (4 x) + \frac{d}{d x} (- 5)$

Evaluate $\frac{d}{d x} [4 x]$ .

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$f'' (x) = 6 x + 4 \frac{d}{d x} (x) + \frac{d}{d x} (- 5)$

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

$f'' (x) = 6 x + 4 \cdot 1 + \frac{d}{d x} (- 5)$

Multiply $4$ by $1$ .

$f'' (x) = 6 x + 4 + \frac{d}{d x} (- 5)$

Differentiate using the Constant Rule.

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$f'' (x) = 6 x + 4 + 0$

Add $6 x + 4$ and $0$ .

$f'' (x) = 6 x + 4$

Step 4

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

$3 x^{2} + 4 x - 5 = 0$

Step 5

Find the first derivative.

Differentiate.

By the Sum Rule, the derivative of $x^{3} + 2 x^{2} - 5 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x]$ .

$f' (x) = \frac{d}{d x} (x^{3}) + \frac{d}{d x} (2 x^{2}) + \frac{d}{d x} (- 5 x)$

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

$f' (x) = 3 x^{2} + \frac{d}{d x} (2 x^{2}) + \frac{d}{d x} (- 5 x)$

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

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

$f' (x) = 3 x^{2} + 2 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 5 x)$

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

$f' (x) = 3 x^{2} + 2 (2 x) + \frac{d}{d x} (- 5 x)$

Multiply $2$ by $2$ .

$f' (x) = 3 x^{2} + 4 x + \frac{d}{d x} (- 5 x)$

Evaluate $\frac{d}{d x} [- 5 x]$ .

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$f' (x) = 3 x^{2} + 4 x - 5 \frac{d}{d x} x$

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

$f' (x) = 3 x^{2} + 4 x - 5 \cdot 1$

Multiply $- 5$ by $1$ .

$f' (x) = 3 x^{2} + 4 x - 5$

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} + 4 x - 5$ .

$3 x^{2} + 4 x - 5$

Step 6

Set the first derivative equal to $0$ then solve the equation $3 x^{2} + 4 x - 5 = 0$ .

Set the first derivative equal to $0$ .

$3 x^{2} + 4 x - 5 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 3$ , $b = 4$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (3 \cdot - 5)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 3 \cdot - 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 4 \pm \sqrt{16 - 12 \cdot - 5}}{2 \cdot 3}$

Multiply $- 12$ by $- 5$ .

$x = \frac{- 4 \pm \sqrt{16 + 60}}{2 \cdot 3}$

Add $16$ and $60$ .

$x = \frac{- 4 \pm \sqrt{76}}{2 \cdot 3}$

Rewrite $76$ as $2^{2} \cdot 19$ .

Factor $4$ out of $76$ .

$x = \frac{- 4 \pm \sqrt{4 (19)}}{2 \cdot 3}$

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{19}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 4 \pm 2 \sqrt{19}}{6}$

Simplify $\frac{- 4 \pm 2 \sqrt{19}}{6}$ .

$x = \frac{- 2 \pm \sqrt{19}}{3}$

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

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 3 \cdot - 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 4 \pm \sqrt{16 - 12 \cdot - 5}}{2 \cdot 3}$

Multiply $- 12$ by $- 5$ .

$x = \frac{- 4 \pm \sqrt{16 + 60}}{2 \cdot 3}$

Add $16$ and $60$ .

$x = \frac{- 4 \pm \sqrt{76}}{2 \cdot 3}$

Rewrite $76$ as $2^{2} \cdot 19$ .

Factor $4$ out of $76$ .

$x = \frac{- 4 \pm \sqrt{4 (19)}}{2 \cdot 3}$

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{19}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 4 \pm 2 \sqrt{19}}{6}$

Simplify $\frac{- 4 \pm 2 \sqrt{19}}{6}$ .

$x = \frac{- 2 \pm \sqrt{19}}{3}$

Change the $\pm$ to $+$ .

$x = \frac{- 2 + \sqrt{19}}{3}$

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 + \sqrt{19}}{3}$

Factor $- 1$ out of $\sqrt{19}$ .

$x = \frac{- 1 \cdot 2 - 1 (- \sqrt{19})}{3}$

Factor $- 1$ out of $- 1 (2) - 1 (- \sqrt{19})$ .

$x = \frac{- 1 (2 - \sqrt{19})}{3}$

Move the negative in front of the fraction.

$x = - \frac{2 - \sqrt{19}}{3}$

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

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 3 \cdot - 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 4 \pm \sqrt{16 - 12 \cdot - 5}}{2 \cdot 3}$

Multiply $- 12$ by $- 5$ .

$x = \frac{- 4 \pm \sqrt{16 + 60}}{2 \cdot 3}$

Add $16$ and $60$ .

$x = \frac{- 4 \pm \sqrt{76}}{2 \cdot 3}$

Rewrite $76$ as $2^{2} \cdot 19$ .

Factor $4$ out of $76$ .

$x = \frac{- 4 \pm \sqrt{4 (19)}}{2 \cdot 3}$

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{19}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 4 \pm 2 \sqrt{19}}{6}$

Simplify $\frac{- 4 \pm 2 \sqrt{19}}{6}$ .

$x = \frac{- 2 \pm \sqrt{19}}{3}$

Change the $\pm$ to $-$ .

$x = \frac{- 2 - \sqrt{19}}{3}$

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 - \sqrt{19}}{3}$

Factor $- 1$ out of $- \sqrt{19}$ .

$x = \frac{- 1 \cdot 2 - (\sqrt{19})}{3}$

Factor $- 1$ out of $- 1 (2) - (\sqrt{19})$ .

$x = \frac{- 1 (2 + \sqrt{19})}{3}$

Move the negative in front of the fraction.

$x = - \frac{2 + \sqrt{19}}{3}$

The final answer is the combination of both solutions.

$x = - \frac{2 - \sqrt{19}}{3}, - \frac{2 + \sqrt{19}}{3}$

Step 7

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 8

Critical points to evaluate.

$x = - \frac{2 - \sqrt{19}}{3}, - \frac{2 + \sqrt{19}}{3}$

Step 9

Evaluate the second derivative at $x = - \frac{2 - \sqrt{19}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$6 (- \frac{2 - \sqrt{19}}{3}) + 4$

Step 10

Evaluate the second derivative.

Simplify each term.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{2 - \sqrt{19}}{3}$ into the numerator.

$6 \frac{- (2 - \sqrt{19})}{3} + 4$

Factor $3$ out of $6$ .

$3 (2) \frac{- (2 - \sqrt{19})}{3} + 4$

Cancel the common factor.

$3 \cdot 2 \frac{- (2 - \sqrt{19})}{3} + 4$

Rewrite the expression.

$2 (- (2 - \sqrt{19})) + 4$

Multiply $- 1$ by $2$ .

$- 2 (2 - \sqrt{19}) + 4$

Apply the distributive property.

$- 2 \cdot 2 - 2 (- \sqrt{19}) + 4$

Multiply $- 2$ by $2$ .

$- 4 - 2 (- \sqrt{19}) + 4$

Multiply $- 1$ by $- 2$ .

$- 4 + 2 \sqrt{19} + 4$

Simplify by adding numbers.

Add $- 4$ and $4$ .

$0 + 2 \sqrt{19}$

Add $0$ and $2 \sqrt{19}$ .

$2 \sqrt{19}$

Step 11

$x = - \frac{2 - \sqrt{19}}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{2 - \sqrt{19}}{3}$ is a local minimum

Step 12

Find the y-value when $x = - \frac{2 - \sqrt{19}}{3}$ .

Replace the variable $x$ with $- \frac{2 - \sqrt{19}}{3}$ in the expression.

$f (- \frac{2 - \sqrt{19}}{3}) = {(- \frac{2 - \sqrt{19}}{3})}^{3} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

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 $- \frac{2 - \sqrt{19}}{3}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = {(- 1)}^{3} {(\frac{2 - \sqrt{19}}{3})}^{3} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Apply the product rule to $\frac{2 - \sqrt{19}}{3}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = {(- 1)}^{3} (\frac{{(2 - \sqrt{19})}^{3}}{3^{3}}) + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{{(2 - \sqrt{19})}^{3}}{3^{3}} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{{(2 - \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Use the Binomial Theorem.

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{2^{3} + 3 \cdot (2^{2} (- \sqrt{19})) + 3 \cdot (2 {(- \sqrt{19})}^{2}) + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Simplify each term.

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 + 3 \cdot (2^{2} (- \sqrt{19})) + 3 \cdot (2 {(- \sqrt{19})}^{2}) + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 + 3 \cdot (4 (- \sqrt{19})) + 3 \cdot (2 {(- \sqrt{19})}^{2}) + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply $3$ by $4$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 + 12 (- \sqrt{19}) + 3 \cdot (2 {(- \sqrt{19})}^{2}) + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply $- 1$ by $12$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 3 \cdot (2 {(- \sqrt{19})}^{2}) + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply $3$ by $2$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 6 {(- \sqrt{19})}^{2} + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Apply the product rule to $- \sqrt{19}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 6 ({(- 1)}^{2} {\sqrt{19}}^{2}) + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 6 (1 {\sqrt{19}}^{2}) + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply ${\sqrt{19}}^{2}$ by $1$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 6 {\sqrt{19}}^{2} + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 6 {(19^{\frac{1}{2}})}^{2} + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 6 \cdot 19^{\frac{1}{2} \cdot 2} + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 6 \cdot 19^{\frac{2}{2}} + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 6 \cdot 19^{\frac{2}{2}} + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Rewrite the expression.

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 6 \cdot 19 + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Evaluate the exponent.

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 6 \cdot 19 + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply $6$ by $19$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 114 + {(- \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Apply the product rule to $- \sqrt{19}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 114 + {(- 1)}^{3} {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 114 - {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Rewrite ${\sqrt{19}}^{3}$ as ${(19^{3})}^{\frac{1}{2}}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 114 - \sqrt{19^{3}}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Raise $19$ to the power of $3$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 114 - \sqrt{6859}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Rewrite $6859$ as $19^{2} \cdot 19$ .

Factor $361$ out of $6859$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 114 - \sqrt{361 (19)}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Rewrite $361$ as $19^{2}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 114 - \sqrt{19^{2} \cdot 19}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Pull terms out from under the radical.

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 114 - (19 \sqrt{19})}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply $19$ by $- 1$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{8 - 12 \sqrt{19} + 114 - 19 \sqrt{19}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Add $8$ and $114$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 12 \sqrt{19} - 19 \sqrt{19}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

Subtract $19 \sqrt{19}$ from $- 12 \sqrt{19}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 {(- \frac{2 - \sqrt{19}}{3})}^{2} - 5 (- \frac{2 - \sqrt{19}}{3})$

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

Apply the product rule to $- \frac{2 - \sqrt{19}}{3}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 ({(- 1)}^{2} {(\frac{2 - \sqrt{19}}{3})}^{2}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Apply the product rule to $\frac{2 - \sqrt{19}}{3}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 ({(- 1)}^{2} (\frac{{(2 - \sqrt{19})}^{2}}{3^{2}})) - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (1 (\frac{{(2 - \sqrt{19})}^{2}}{3^{2}})) - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply $\frac{{(2 - \sqrt{19})}^{2}}{3^{2}}$ by $1$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{{(2 - \sqrt{19})}^{2}}{3^{2}}) - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{{(2 - \sqrt{19})}^{2}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Rewrite ${(2 - \sqrt{19})}^{2}$ as $(2 - \sqrt{19}) (2 - \sqrt{19})$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{(2 - \sqrt{19}) (2 - \sqrt{19})}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Expand $(2 - \sqrt{19}) (2 - \sqrt{19})$ using the FOIL Method.

Apply the distributive property.

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{2 (2 - \sqrt{19}) - \sqrt{19} (2 - \sqrt{19})}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Apply the distributive property.

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{2 \cdot 2 + 2 (- \sqrt{19}) - \sqrt{19} (2 - \sqrt{19})}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Apply the distributive property.

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{2 \cdot 2 + 2 (- \sqrt{19}) - \sqrt{19} \cdot 2 - \sqrt{19} (- \sqrt{19})}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 + 2 (- \sqrt{19}) - \sqrt{19} \cdot 2 - \sqrt{19} (- \sqrt{19})}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply $- 1$ by $2$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - \sqrt{19} \cdot 2 - \sqrt{19} (- \sqrt{19})}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply $2$ by $- 1$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} - \sqrt{19} (- \sqrt{19})}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply $- \sqrt{19} (- \sqrt{19})$ .

Multiply $- 1$ by $- 1$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + 1 \sqrt{19} \sqrt{19}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply $\sqrt{19}$ by $1$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + \sqrt{19} \sqrt{19}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Raise $\sqrt{19}$ to the power of $1$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + \sqrt{19} \sqrt{19}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Raise $\sqrt{19}$ to the power of $1$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + \sqrt{19} \sqrt{19}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + {\sqrt{19}}^{1 + 1}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Add $1$ and $1$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + {\sqrt{19}}^{2}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + {(19^{\frac{1}{2}})}^{2}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + 19^{\frac{1}{2} \cdot 2}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

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

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + 19^{\frac{2}{2}}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + 19^{\frac{2}{2}}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Rewrite the expression.

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + 19}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Evaluate the exponent.

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{4 - 2 \sqrt{19} - 2 \sqrt{19} + 19}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Add $4$ and $19$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{23 - 2 \sqrt{19} - 2 \sqrt{19}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Subtract $2 \sqrt{19}$ from $- 2 \sqrt{19}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + 2 (\frac{23 - 4 \sqrt{19}}{9}) - 5 (- \frac{2 - \sqrt{19}}{3})$

Combine $2$ and $\frac{23 - 4 \sqrt{19}}{9}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + \frac{2 (23 - 4 \sqrt{19})}{9} - 5 (- \frac{2 - \sqrt{19}}{3})$

Multiply $- 5 (- \frac{2 - \sqrt{19}}{3})$ .

Multiply $- 1$ by $- 5$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + \frac{2 (23 - 4 \sqrt{19})}{9} + 5 (\frac{2 - \sqrt{19}}{3})$

Combine $5$ and $\frac{2 - \sqrt{19}}{3}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + \frac{2 (23 - 4 \sqrt{19})}{9} + \frac{5 (2 - \sqrt{19})}{3}$

Find the common denominator.

Multiply $\frac{2 (23 - 4 \sqrt{19})}{9}$ by $\frac{3}{3}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + \frac{2 (23 - 4 \sqrt{19})}{9} \cdot \frac{3}{3} + \frac{5 (2 - \sqrt{19})}{3}$

Multiply $\frac{2 (23 - 4 \sqrt{19})}{9}$ by $\frac{3}{3}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + \frac{2 (23 - 4 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{5 (2 - \sqrt{19})}{3}$

Multiply $\frac{5 (2 - \sqrt{19})}{3}$ by $\frac{9}{9}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + \frac{2 (23 - 4 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{5 (2 - \sqrt{19})}{3} \cdot \frac{9}{9}$

Multiply $\frac{5 (2 - \sqrt{19})}{3}$ by $\frac{9}{9}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + \frac{2 (23 - 4 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{5 (2 - \sqrt{19}) \cdot 9}{3 \cdot 9}$

Reorder the factors of $9 \cdot 3$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + \frac{2 (23 - 4 \sqrt{19}) \cdot 3}{3 \cdot 9} + \frac{5 (2 - \sqrt{19}) \cdot 9}{3 \cdot 9}$

Multiply $3$ by $9$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + \frac{2 (23 - 4 \sqrt{19}) \cdot 3}{27} + \frac{5 (2 - \sqrt{19}) \cdot 9}{3 \cdot 9}$

Multiply $3$ by $9$ .

$f (- \frac{2 - \sqrt{19}}{3}) = - \frac{122 - 31 \sqrt{19}}{27} + \frac{2 (23 - 4 \sqrt{19}) \cdot 3}{27} + \frac{5 (2 - \sqrt{19}) \cdot 9}{27}$

Combine the numerators over the common denominator.

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- (122 - 31 \sqrt{19}) + 2 (23 - 4 \sqrt{19}) \cdot 3 + 5 (2 - \sqrt{19}) \cdot 9}{27}$

Simplify each term.

Apply the distributive property.

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 1 \cdot 122 - (- 31 \sqrt{19}) + 2 (23 - 4 \sqrt{19}) \cdot 3 + 5 (2 - \sqrt{19}) \cdot 9}{27}$

Multiply $- 1$ by $122$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 - (- 31 \sqrt{19}) + 2 (23 - 4 \sqrt{19}) \cdot 3 + 5 (2 - \sqrt{19}) \cdot 9}{27}$

Multiply $- 31$ by $- 1$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + 2 (23 - 4 \sqrt{19}) \cdot 3 + 5 (2 - \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + (2 \cdot 23 + 2 (- 4 \sqrt{19})) \cdot 3 + 5 (2 - \sqrt{19}) \cdot 9}{27}$

Multiply $2$ by $23$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + (46 + 2 (- 4 \sqrt{19})) \cdot 3 + 5 (2 - \sqrt{19}) \cdot 9}{27}$

Multiply $- 4$ by $2$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + (46 - 8 \sqrt{19}) \cdot 3 + 5 (2 - \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + 46 \cdot 3 - 8 \sqrt{19} \cdot 3 + 5 (2 - \sqrt{19}) \cdot 9}{27}$

Multiply $46$ by $3$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + 138 - 8 \sqrt{19} \cdot 3 + 5 (2 - \sqrt{19}) \cdot 9}{27}$

Multiply $3$ by $- 8$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + 138 - 24 \sqrt{19} + 5 (2 - \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + 138 - 24 \sqrt{19} + (5 \cdot 2 + 5 (- \sqrt{19})) \cdot 9}{27}$

Multiply $5$ by $2$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + 138 - 24 \sqrt{19} + (10 + 5 (- \sqrt{19})) \cdot 9}{27}$

Multiply $- 1$ by $5$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + 138 - 24 \sqrt{19} + (10 - 5 \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + 138 - 24 \sqrt{19} + 10 \cdot 9 - 5 \sqrt{19} \cdot 9}{27}$

Multiply $10$ by $9$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + 138 - 24 \sqrt{19} + 90 - 5 \sqrt{19} \cdot 9}{27}$

Multiply $9$ by $- 5$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{- 122 + 31 \sqrt{19} + 138 - 24 \sqrt{19} + 90 - 45 \sqrt{19}}{27}$

Simplify by adding terms.

Add $- 122$ and $138$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{16 + 31 \sqrt{19} - 24 \sqrt{19} + 90 - 45 \sqrt{19}}{27}$

Add $16$ and $90$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{106 + 31 \sqrt{19} - 24 \sqrt{19} - 45 \sqrt{19}}{27}$

Subtract $24 \sqrt{19}$ from $31 \sqrt{19}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{106 + 7 \sqrt{19} - 45 \sqrt{19}}{27}$

Subtract $45 \sqrt{19}$ from $7 \sqrt{19}$ .

$f (- \frac{2 - \sqrt{19}}{3}) = \frac{106 - 38 \sqrt{19}}{27}$

The final answer is $\frac{106 - 38 \sqrt{19}}{27}$ .

$y = \frac{106 - 38 \sqrt{19}}{27}$

Step 13

Evaluate the second derivative at $x = - \frac{2 + \sqrt{19}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$6 (- \frac{2 + \sqrt{19}}{3}) + 4$

Step 14

Evaluate the second derivative.

Simplify each term.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{2 + \sqrt{19}}{3}$ into the numerator.

$6 \frac{- (2 + \sqrt{19})}{3} + 4$

Factor $3$ out of $6$ .

$3 (2) \frac{- (2 + \sqrt{19})}{3} + 4$

Cancel the common factor.

$3 \cdot 2 \frac{- (2 + \sqrt{19})}{3} + 4$

Rewrite the expression.

$2 (- (2 + \sqrt{19})) + 4$

Multiply $- 1$ by $2$ .

$- 2 (2 + \sqrt{19}) + 4$

Apply the distributive property.

$- 2 \cdot 2 - 2 \sqrt{19} + 4$

Multiply $- 2$ by $2$ .

$- 4 - 2 \sqrt{19} + 4$

Simplify by adding numbers.

Add $- 4$ and $4$ .

$0 - 2 \sqrt{19}$

Subtract $2 \sqrt{19}$ from $0$ .

$- 2 \sqrt{19}$

Step 15

$x = - \frac{2 + \sqrt{19}}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{2 + \sqrt{19}}{3}$ is a local maximum

Step 16

Find the y-value when $x = - \frac{2 + \sqrt{19}}{3}$ .

Replace the variable $x$ with $- \frac{2 + \sqrt{19}}{3}$ in the expression.

$f (- \frac{2 + \sqrt{19}}{3}) = {(- \frac{2 + \sqrt{19}}{3})}^{3} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

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 $- \frac{2 + \sqrt{19}}{3}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = {(- 1)}^{3} {(\frac{2 + \sqrt{19}}{3})}^{3} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Apply the product rule to $\frac{2 + \sqrt{19}}{3}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = {(- 1)}^{3} (\frac{{(2 + \sqrt{19})}^{3}}{3^{3}}) + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

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

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{{(2 + \sqrt{19})}^{3}}{3^{3}} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

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

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{{(2 + \sqrt{19})}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Use the Binomial Theorem.

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{2^{3} + 3 \cdot (2^{2} \sqrt{19}) + 3 \cdot (2 {\sqrt{19}}^{2}) + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Simplify each term.

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

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 3 \cdot (2^{2} \sqrt{19}) + 3 \cdot (2 {\sqrt{19}}^{2}) + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

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

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 3 \cdot (4 \sqrt{19}) + 3 \cdot (2 {\sqrt{19}}^{2}) + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Multiply $3$ by $4$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 3 \cdot (2 {\sqrt{19}}^{2}) + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Multiply $3$ by $2$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 6 {\sqrt{19}}^{2} + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 6 {(19^{\frac{1}{2}})}^{2} + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

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

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 6 \cdot 19^{\frac{1}{2} \cdot 2} + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

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

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 6 \cdot 19^{\frac{2}{2}} + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 6 \cdot 19^{\frac{2}{2}} + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Rewrite the expression.

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 6 \cdot 19 + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Evaluate the exponent.

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 6 \cdot 19 + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Multiply $6$ by $19$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 114 + {\sqrt{19}}^{3}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Rewrite ${\sqrt{19}}^{3}$ as ${(19^{3})}^{\frac{1}{2}}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 114 + \sqrt{19^{3}}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Raise $19$ to the power of $3$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 114 + \sqrt{6859}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Rewrite $6859$ as $19^{2} \cdot 19$ .

Factor $361$ out of $6859$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 114 + \sqrt{361 (19)}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Rewrite $361$ as $19^{2}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 114 + \sqrt{19^{2} \cdot 19}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Pull terms out from under the radical.

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{8 + 12 \sqrt{19} + 114 + 19 \sqrt{19}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Add $8$ and $114$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 12 \sqrt{19} + 19 \sqrt{19}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

Add $12 \sqrt{19}$ and $19 \sqrt{19}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 {(- \frac{2 + \sqrt{19}}{3})}^{2} - 5 (- \frac{2 + \sqrt{19}}{3})$

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

Apply the product rule to $- \frac{2 + \sqrt{19}}{3}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 ({(- 1)}^{2} {(\frac{2 + \sqrt{19}}{3})}^{2}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Apply the product rule to $\frac{2 + \sqrt{19}}{3}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 ({(- 1)}^{2} (\frac{{(2 + \sqrt{19})}^{2}}{3^{2}})) - 5 (- \frac{2 + \sqrt{19}}{3})$

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

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (1 (\frac{{(2 + \sqrt{19})}^{2}}{3^{2}})) - 5 (- \frac{2 + \sqrt{19}}{3})$

Multiply $\frac{{(2 + \sqrt{19})}^{2}}{3^{2}}$ by $1$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{{(2 + \sqrt{19})}^{2}}{3^{2}}) - 5 (- \frac{2 + \sqrt{19}}{3})$

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

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{{(2 + \sqrt{19})}^{2}}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Rewrite ${(2 + \sqrt{19})}^{2}$ as $(2 + \sqrt{19}) (2 + \sqrt{19})$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{(2 + \sqrt{19}) (2 + \sqrt{19})}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Expand $(2 + \sqrt{19}) (2 + \sqrt{19})$ using the FOIL Method.

Apply the distributive property.

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{2 (2 + \sqrt{19}) + \sqrt{19} (2 + \sqrt{19})}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Apply the distributive property.

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{2 \cdot 2 + 2 \sqrt{19} + \sqrt{19} (2 + \sqrt{19})}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Apply the distributive property.

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{2 \cdot 2 + 2 \sqrt{19} + \sqrt{19} \cdot 2 + \sqrt{19} \sqrt{19}}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{4 + 2 \sqrt{19} + \sqrt{19} \cdot 2 + \sqrt{19} \sqrt{19}}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Move $2$ to the left of $\sqrt{19}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{4 + 2 \sqrt{19} + 2 \cdot \sqrt{19} + \sqrt{19} \sqrt{19}}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Combine using the product rule for radicals.

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{4 + 2 \sqrt{19} + 2 \sqrt{19} + \sqrt{19 \cdot 19}}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Multiply $19$ by $19$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{4 + 2 \sqrt{19} + 2 \sqrt{19} + \sqrt{361}}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Rewrite $361$ as $19^{2}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{4 + 2 \sqrt{19} + 2 \sqrt{19} + \sqrt{19^{2}}}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Pull terms out from under the radical, assuming positive real numbers.

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{4 + 2 \sqrt{19} + 2 \sqrt{19} + 19}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Add $4$ and $19$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{23 + 2 \sqrt{19} + 2 \sqrt{19}}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Add $2 \sqrt{19}$ and $2 \sqrt{19}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + 2 (\frac{23 + 4 \sqrt{19}}{9}) - 5 (- \frac{2 + \sqrt{19}}{3})$

Combine $2$ and $\frac{23 + 4 \sqrt{19}}{9}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + \frac{2 (23 + 4 \sqrt{19})}{9} - 5 (- \frac{2 + \sqrt{19}}{3})$

Multiply $- 5 (- \frac{2 + \sqrt{19}}{3})$ .

Multiply $- 1$ by $- 5$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + \frac{2 (23 + 4 \sqrt{19})}{9} + 5 (\frac{2 + \sqrt{19}}{3})$

Combine $5$ and $\frac{2 + \sqrt{19}}{3}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + \frac{2 (23 + 4 \sqrt{19})}{9} + \frac{5 (2 + \sqrt{19})}{3}$

Find the common denominator.

Multiply $\frac{2 (23 + 4 \sqrt{19})}{9}$ by $\frac{3}{3}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + \frac{2 (23 + 4 \sqrt{19})}{9} \cdot \frac{3}{3} + \frac{5 (2 + \sqrt{19})}{3}$

Multiply $\frac{2 (23 + 4 \sqrt{19})}{9}$ by $\frac{3}{3}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + \frac{2 (23 + 4 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{5 (2 + \sqrt{19})}{3}$

Multiply $\frac{5 (2 + \sqrt{19})}{3}$ by $\frac{9}{9}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + \frac{2 (23 + 4 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{5 (2 + \sqrt{19})}{3} \cdot \frac{9}{9}$

Multiply $\frac{5 (2 + \sqrt{19})}{3}$ by $\frac{9}{9}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + \frac{2 (23 + 4 \sqrt{19}) \cdot 3}{9 \cdot 3} + \frac{5 (2 + \sqrt{19}) \cdot 9}{3 \cdot 9}$

Reorder the factors of $9 \cdot 3$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + \frac{2 (23 + 4 \sqrt{19}) \cdot 3}{3 \cdot 9} + \frac{5 (2 + \sqrt{19}) \cdot 9}{3 \cdot 9}$

Multiply $3$ by $9$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + \frac{2 (23 + 4 \sqrt{19}) \cdot 3}{27} + \frac{5 (2 + \sqrt{19}) \cdot 9}{3 \cdot 9}$

Multiply $3$ by $9$ .

$f (- \frac{2 + \sqrt{19}}{3}) = - \frac{122 + 31 \sqrt{19}}{27} + \frac{2 (23 + 4 \sqrt{19}) \cdot 3}{27} + \frac{5 (2 + \sqrt{19}) \cdot 9}{27}$

Combine the numerators over the common denominator.

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- (122 + 31 \sqrt{19}) + 2 (23 + 4 \sqrt{19}) \cdot 3 + 5 (2 + \sqrt{19}) \cdot 9}{27}$

Simplify each term.

Apply the distributive property.

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 1 \cdot 122 - (31 \sqrt{19}) + 2 (23 + 4 \sqrt{19}) \cdot 3 + 5 (2 + \sqrt{19}) \cdot 9}{27}$

Multiply $- 1$ by $122$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - (31 \sqrt{19}) + 2 (23 + 4 \sqrt{19}) \cdot 3 + 5 (2 + \sqrt{19}) \cdot 9}{27}$

Multiply $31$ by $- 1$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + 2 (23 + 4 \sqrt{19}) \cdot 3 + 5 (2 + \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + (2 \cdot 23 + 2 (4 \sqrt{19})) \cdot 3 + 5 (2 + \sqrt{19}) \cdot 9}{27}$

Multiply $2$ by $23$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + (46 + 2 (4 \sqrt{19})) \cdot 3 + 5 (2 + \sqrt{19}) \cdot 9}{27}$

Multiply $4$ by $2$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + (46 + 8 \sqrt{19}) \cdot 3 + 5 (2 + \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + 46 \cdot 3 + 8 \sqrt{19} \cdot 3 + 5 (2 + \sqrt{19}) \cdot 9}{27}$

Multiply $46$ by $3$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + 138 + 8 \sqrt{19} \cdot 3 + 5 (2 + \sqrt{19}) \cdot 9}{27}$

Multiply $3$ by $8$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + 138 + 24 \sqrt{19} + 5 (2 + \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + 138 + 24 \sqrt{19} + (5 \cdot 2 + 5 \sqrt{19}) \cdot 9}{27}$

Multiply $5$ by $2$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + 138 + 24 \sqrt{19} + (10 + 5 \sqrt{19}) \cdot 9}{27}$

Apply the distributive property.

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + 138 + 24 \sqrt{19} + 10 \cdot 9 + 5 \sqrt{19} \cdot 9}{27}$

Multiply $10$ by $9$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + 138 + 24 \sqrt{19} + 90 + 5 \sqrt{19} \cdot 9}{27}$

Multiply $9$ by $5$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{- 122 - 31 \sqrt{19} + 138 + 24 \sqrt{19} + 90 + 45 \sqrt{19}}{27}$

Simplify by adding terms.

Add $- 122$ and $138$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{16 - 31 \sqrt{19} + 24 \sqrt{19} + 90 + 45 \sqrt{19}}{27}$

Add $16$ and $90$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{106 - 31 \sqrt{19} + 24 \sqrt{19} + 45 \sqrt{19}}{27}$

Add $- 31 \sqrt{19}$ and $24 \sqrt{19}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{106 - 7 \sqrt{19} + 45 \sqrt{19}}{27}$

Add $- 7 \sqrt{19}$ and $45 \sqrt{19}$ .

$f (- \frac{2 + \sqrt{19}}{3}) = \frac{106 + 38 \sqrt{19}}{27}$

The final answer is $\frac{106 + 38 \sqrt{19}}{27}$ .

$y = \frac{106 + 38 \sqrt{19}}{27}$

Step 17

These are the local extrema for $f (x) = x^{3} + 2 x^{2} - 5 x$ .

$(- \frac{2 - \sqrt{19}}{3}, \frac{106 - 38 \sqrt{19}}{27})$ is a local minima

$(- \frac{2 + \sqrt{19}}{3}, \frac{106 + 38 \sqrt{19}}{27})$ is a local maxima

Step 18

Find the Local Maxima and Minima x^3+2x^2-5x

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