Find the Local Maxima and Minima y=3x^3+15x^2-12x-60

$y = 3 x^{3} + 15 x^{2} - 12 x - 60$

Step 1

Write $y = 3 x^{3} + 15 x^{2} - 12 x - 60$ as a function.

$f (x) = 3 x^{3} + 15 x^{2} - 12 x - 60$

Step 2

Find the first derivative of the function.

By the Sum Rule, the derivative of $3 x^{3} + 15 x^{2} - 12 x - 60$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 60]$ .

$\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 60]$

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

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

$3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 60]$

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

$3 (3 x^{2}) + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 60]$

Multiply $3$ by $3$ .

$9 x^{2} + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 60]$

Evaluate $\frac{d}{d x} [15 x^{2}]$ .

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{2}$ with respect to $x$ is $15 \frac{d}{d x} [x^{2}]$ .

$9 x^{2} + 15 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 60]$

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

$9 x^{2} + 15 (2 x) + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 60]$

Multiply $2$ by $15$ .

$9 x^{2} + 30 x + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 60]$

Evaluate $\frac{d}{d x} [- 12 x]$ .

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$9 x^{2} + 30 x - 12 \frac{d}{d x} [x] + \frac{d}{d x} [- 60]$

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

$9 x^{2} + 30 x - 12 \cdot 1 + \frac{d}{d x} [- 60]$

Multiply $- 12$ by $1$ .

$9 x^{2} + 30 x - 12 + \frac{d}{d x} [- 60]$

Differentiate using the Constant Rule.

Since $- 60$ is constant with respect to $x$ , the derivative of $- 60$ with respect to $x$ is $0$ .

$9 x^{2} + 30 x - 12 + 0$

Add $9 x^{2} + 30 x - 12$ and $0$ .

$9 x^{2} + 30 x - 12$

Step 3

Find the second derivative of the function.

By the Sum Rule, the derivative of $9 x^{2} + 30 x - 12$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [30 x] + \frac{d}{d x} [- 12]$ .

$f'' (x) = \frac{d}{d x} (9 x^{2}) + \frac{d}{d x} (30 x) + \frac{d}{d x} (- 12)$

Evaluate $\frac{d}{d x} [9 x^{2}]$ .

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 9 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (30 x) + \frac{d}{d x} (- 12)$

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

$f'' (x) = 9 (2 x) + \frac{d}{d x} (30 x) + \frac{d}{d x} (- 12)$

Multiply $2$ by $9$ .

$f'' (x) = 18 x + \frac{d}{d x} (30 x) + \frac{d}{d x} (- 12)$

Evaluate $\frac{d}{d x} [30 x]$ .

Since $30$ is constant with respect to $x$ , the derivative of $30 x$ with respect to $x$ is $30 \frac{d}{d x} [x]$ .

$f'' (x) = 18 x + 30 \frac{d}{d x} (x) + \frac{d}{d x} (- 12)$

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

$f'' (x) = 18 x + 30 \cdot 1 + \frac{d}{d x} (- 12)$

Multiply $30$ by $1$ .

$f'' (x) = 18 x + 30 + \frac{d}{d x} (- 12)$

Differentiate using the Constant Rule.

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$f'' (x) = 18 x + 30 + 0$

Add $18 x + 30$ and $0$ .

$f'' (x) = 18 x + 30$

Step 4

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

$9 x^{2} + 30 x - 12 = 0$

Step 5

Find the first derivative.

By the Sum Rule, the derivative of $3 x^{3} + 15 x^{2} - 12 x - 60$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 60]$ .

$f' (x) = \frac{d}{d x} (3 x^{3}) + \frac{d}{d x} (15 x^{2}) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (- 60)$

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

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

$f' (x) = 3 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (15 x^{2}) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (- 60)$

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

$f' (x) = 3 (3 x^{2}) + \frac{d}{d x} (15 x^{2}) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (- 60)$

Multiply $3$ by $3$ .

$f' (x) = 9 x^{2} + \frac{d}{d x} (15 x^{2}) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (- 60)$

Evaluate $\frac{d}{d x} [15 x^{2}]$ .

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{2}$ with respect to $x$ is $15 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 9 x^{2} + 15 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (- 60)$

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

$f' (x) = 9 x^{2} + 15 (2 x) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (- 60)$

Multiply $2$ by $15$ .

$f' (x) = 9 x^{2} + 30 x + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (- 60)$

Evaluate $\frac{d}{d x} [- 12 x]$ .

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$f' (x) = 9 x^{2} + 30 x - 12 \frac{d}{d x} x + \frac{d}{d x} (- 60)$

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

$f' (x) = 9 x^{2} + 30 x - 12 \cdot 1 + \frac{d}{d x} (- 60)$

Multiply $- 12$ by $1$ .

$f' (x) = 9 x^{2} + 30 x - 12 + \frac{d}{d x} (- 60)$

Differentiate using the Constant Rule.

Since $- 60$ is constant with respect to $x$ , the derivative of $- 60$ with respect to $x$ is $0$ .

$f' (x) = 9 x^{2} + 30 x - 12 + 0$

Add $9 x^{2} + 30 x - 12$ and $0$ .

$f' (x) = 9 x^{2} + 30 x - 12$

The first derivative of $f (x)$ with respect to $x$ is $9 x^{2} + 30 x - 12$ .

$9 x^{2} + 30 x - 12$

Step 6

Set the first derivative equal to $0$ then solve the equation $9 x^{2} + 30 x - 12 = 0$ .

Set the first derivative equal to $0$ .

$9 x^{2} + 30 x - 12 = 0$

Factor $3$ out of $9 x^{2} + 30 x - 12$ .

Factor $3$ out of $9 x^{2}$ .

$3 (3 x^{2}) + 30 x - 12 = 0$

Factor $3$ out of $30 x$ .

$3 (3 x^{2}) + 3 (10 x) - 12 = 0$

Factor $3$ out of $- 12$ .

$3 (3 x^{2}) + 3 (10 x) + 3 (- 4) = 0$

Factor $3$ out of $3 (3 x^{2}) + 3 (10 x)$ .

$3 (3 x^{2} + 10 x) + 3 (- 4) = 0$

Factor $3$ out of $3 (3 x^{2} + 10 x) + 3 (- 4)$ .

$3 (3 x^{2} + 10 x - 4) = 0$

Divide each term in $3 (3 x^{2} + 10 x - 4) = 0$ by $3$ and simplify.

Divide each term in $3 (3 x^{2} + 10 x - 4) = 0$ by $3$ .

$\frac{3 (3 x^{2} + 10 x - 4)}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (3 x^{2} + 10 x - 4)}{3} = \frac{0}{3}$

Divide $3 x^{2} + 10 x - 4$ by $1$ .

$3 x^{2} + 10 x - 4 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$3 x^{2} + 10 x - 4 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 3$ , $b = 10$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (3 \cdot - 4)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

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

Multiply $- 4$ by $3$ .

$x = \frac{- 10 \pm \sqrt{100 - 12 \cdot - 4}}{2 \cdot 3}$

Multiply $- 12$ by $- 4$ .

$x = \frac{- 10 \pm \sqrt{100 + 48}}{2 \cdot 3}$

Add $100$ and $48$ .

$x = \frac{- 10 \pm \sqrt{148}}{2 \cdot 3}$

Rewrite $148$ as $2^{2} \cdot 37$ .

Factor $4$ out of $148$ .

$x = \frac{- 10 \pm \sqrt{4 (37)}}{2 \cdot 3}$

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

$x = \frac{- 10 \pm \sqrt{2^{2} \cdot 37}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{- 10 \pm 2 \sqrt{37}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 10 \pm 2 \sqrt{37}}{6}$

Simplify $\frac{- 10 \pm 2 \sqrt{37}}{6}$ .

$x = \frac{- 5 \pm \sqrt{37}}{3}$

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

Simplify the numerator.

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

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

Multiply $- 4$ by $3$ .

$x = \frac{- 10 \pm \sqrt{100 - 12 \cdot - 4}}{2 \cdot 3}$

Multiply $- 12$ by $- 4$ .

$x = \frac{- 10 \pm \sqrt{100 + 48}}{2 \cdot 3}$

Add $100$ and $48$ .

$x = \frac{- 10 \pm \sqrt{148}}{2 \cdot 3}$

Rewrite $148$ as $2^{2} \cdot 37$ .

Factor $4$ out of $148$ .

$x = \frac{- 10 \pm \sqrt{4 (37)}}{2 \cdot 3}$

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

$x = \frac{- 10 \pm \sqrt{2^{2} \cdot 37}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{- 10 \pm 2 \sqrt{37}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 10 \pm 2 \sqrt{37}}{6}$

Simplify $\frac{- 10 \pm 2 \sqrt{37}}{6}$ .

$x = \frac{- 5 \pm \sqrt{37}}{3}$

Change the $\pm$ to $+$ .

$x = \frac{- 5 + \sqrt{37}}{3}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + \sqrt{37}}{3}$

Factor $- 1$ out of $\sqrt{37}$ .

$x = \frac{- 1 \cdot 5 - 1 (- \sqrt{37})}{3}$

Factor $- 1$ out of $- 1 (5) - 1 (- \sqrt{37})$ .

$x = \frac{- 1 (5 - \sqrt{37})}{3}$

Move the negative in front of the fraction.

$x = - \frac{5 - \sqrt{37}}{3}$

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

Simplify the numerator.

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

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

Multiply $- 4$ by $3$ .

$x = \frac{- 10 \pm \sqrt{100 - 12 \cdot - 4}}{2 \cdot 3}$

Multiply $- 12$ by $- 4$ .

$x = \frac{- 10 \pm \sqrt{100 + 48}}{2 \cdot 3}$

Add $100$ and $48$ .

$x = \frac{- 10 \pm \sqrt{148}}{2 \cdot 3}$

Rewrite $148$ as $2^{2} \cdot 37$ .

Factor $4$ out of $148$ .

$x = \frac{- 10 \pm \sqrt{4 (37)}}{2 \cdot 3}$

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

$x = \frac{- 10 \pm \sqrt{2^{2} \cdot 37}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{- 10 \pm 2 \sqrt{37}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 10 \pm 2 \sqrt{37}}{6}$

Simplify $\frac{- 10 \pm 2 \sqrt{37}}{6}$ .

$x = \frac{- 5 \pm \sqrt{37}}{3}$

Change the $\pm$ to $-$ .

$x = \frac{- 5 - \sqrt{37}}{3}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - \sqrt{37}}{3}$

Factor $- 1$ out of $- \sqrt{37}$ .

$x = \frac{- 1 \cdot 5 - (\sqrt{37})}{3}$

Factor $- 1$ out of $- 1 (5) - (\sqrt{37})$ .

$x = \frac{- 1 (5 + \sqrt{37})}{3}$

Move the negative in front of the fraction.

$x = - \frac{5 + \sqrt{37}}{3}$

The final answer is the combination of both solutions.

$x = - \frac{5 - \sqrt{37}}{3}, - \frac{5 + \sqrt{37}}{3}$

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.

$x = - \frac{5 - \sqrt{37}}{3}, - \frac{5 + \sqrt{37}}{3}$

Step 9

Evaluate the second derivative at $x = - \frac{5 - \sqrt{37}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$18 (- \frac{5 - \sqrt{37}}{3}) + 30$

Step 10

Evaluate the second derivative.

Simplify each term.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{5 - \sqrt{37}}{3}$ into the numerator.

$18 \frac{- (5 - \sqrt{37})}{3} + 30$

Factor $3$ out of $18$ .

$3 (6) \frac{- (5 - \sqrt{37})}{3} + 30$

Cancel the common factor.

$3 \cdot 6 \frac{- (5 - \sqrt{37})}{3} + 30$

Rewrite the expression.

$6 (- (5 - \sqrt{37})) + 30$

Multiply $- 1$ by $6$ .

$- 6 (5 - \sqrt{37}) + 30$

Apply the distributive property.

$- 6 \cdot 5 - 6 (- \sqrt{37}) + 30$

Multiply $- 6$ by $5$ .

$- 30 - 6 (- \sqrt{37}) + 30$

Multiply $- 1$ by $- 6$ .

$- 30 + 6 \sqrt{37} + 30$

Simplify by adding numbers.

Add $- 30$ and $30$ .

$0 + 6 \sqrt{37}$

Add $0$ and $6 \sqrt{37}$ .

$6 \sqrt{37}$

Step 11

$x = - \frac{5 - \sqrt{37}}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{5 - \sqrt{37}}{3}$ is a local minimum

Step 12

Find the y-value when $x = - \frac{5 - \sqrt{37}}{3}$ .

Replace the variable $x$ with $- \frac{5 - \sqrt{37}}{3}$ in the expression.

$f (- \frac{5 - \sqrt{37}}{3}) = 3 {(- \frac{5 - \sqrt{37}}{3})}^{3} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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 $- \frac{5 - \sqrt{37}}{3}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = 3 ({(- 1)}^{3} {(\frac{5 - \sqrt{37}}{3})}^{3}) + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Apply the product rule to $\frac{5 - \sqrt{37}}{3}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = 3 ({(- 1)}^{3} (\frac{{(5 - \sqrt{37})}^{3}}{3^{3}})) + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 - \sqrt{37}}{3}) = 3 (- \frac{{(5 - \sqrt{37})}^{3}}{3^{3}}) + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 - \sqrt{37}}{3}) = 3 (- \frac{{(5 - \sqrt{37})}^{3}}{27}) + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{{(5 - \sqrt{37})}^{3}}{27}$ into the numerator.

$f (- \frac{5 - \sqrt{37}}{3}) = 3 (\frac{- {(5 - \sqrt{37})}^{3}}{27}) + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Factor $3$ out of $27$ .

$f (- \frac{5 - \sqrt{37}}{3}) = 3 (\frac{- {(5 - \sqrt{37})}^{3}}{3 (9)}) + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Cancel the common factor.

$f (- \frac{5 - \sqrt{37}}{3}) = 3 (\frac{- {(5 - \sqrt{37})}^{3}}{3 \cdot 9}) + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Rewrite the expression.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- {(5 - \sqrt{37})}^{3}}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Use the Binomial Theorem.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (5^{3} + 3 \cdot (5^{2} (- \sqrt{37})) + 3 \cdot (5 {(- \sqrt{37})}^{2}) + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Simplify each term.

Raise $5$ to the power of $3$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 + 3 \cdot (5^{2} (- \sqrt{37})) + 3 \cdot (5 {(- \sqrt{37})}^{2}) + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Raise $5$ to the power of $2$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 + 3 \cdot (25 (- \sqrt{37})) + 3 \cdot (5 {(- \sqrt{37})}^{2}) + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Multiply $3$ by $25$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 + 75 (- \sqrt{37}) + 3 \cdot (5 {(- \sqrt{37})}^{2}) + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Multiply $- 1$ by $75$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 3 \cdot (5 {(- \sqrt{37})}^{2}) + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Multiply $3$ by $5$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 15 {(- \sqrt{37})}^{2} + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Apply the product rule to $- \sqrt{37}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 15 ({(- 1)}^{2} {\sqrt{37}}^{2}) + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 15 (1 {\sqrt{37}}^{2}) + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Multiply ${\sqrt{37}}^{2}$ by $1$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 15 {\sqrt{37}}^{2} + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Rewrite ${\sqrt{37}}^{2}$ as $37$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{37}$ as $37^{\frac{1}{2}}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 15 {(37^{\frac{1}{2}})}^{2} + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 15 \cdot 37^{\frac{1}{2} \cdot 2} + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 15 \cdot 37^{\frac{2}{2}} + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 15 \cdot 37^{\frac{2}{2}} + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Rewrite the expression.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 15 \cdot 37 + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Evaluate the exponent.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 15 \cdot 37 + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Multiply $15$ by $37$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 555 + {(- \sqrt{37})}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Apply the product rule to $- \sqrt{37}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 555 + {(- 1)}^{3} {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 555 - {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Rewrite ${\sqrt{37}}^{3}$ as ${(37^{3})}^{\frac{1}{2}}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 555 - \sqrt{37^{3}})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Raise $37$ to the power of $3$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 555 - \sqrt{50653})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Rewrite $50653$ as $37^{2} \cdot 37$ .

Factor $1369$ out of $50653$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 555 - \sqrt{1369 (37)})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Rewrite $1369$ as $37^{2}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 555 - \sqrt{37^{2} \cdot 37})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Pull terms out from under the radical.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 555 - (37 \sqrt{37}))}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Multiply $37$ by $- 1$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (125 - 75 \sqrt{37} + 555 - 37 \sqrt{37})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Add $125$ and $555$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (680 - 75 \sqrt{37} - 37 \sqrt{37})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Subtract $37 \sqrt{37}$ from $- 75 \sqrt{37}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (680 - 112 \sqrt{37})}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Move the negative in front of the fraction.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + 15 {(- \frac{5 - \sqrt{37}}{3})}^{2} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

Apply the product rule to $- \frac{5 - \sqrt{37}}{3}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + 15 ({(- 1)}^{2} {(\frac{5 - \sqrt{37}}{3})}^{2}) - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Apply the product rule to $\frac{5 - \sqrt{37}}{3}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + 15 ({(- 1)}^{2} (\frac{{(5 - \sqrt{37})}^{2}}{3^{2}})) - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + 15 (1 (\frac{{(5 - \sqrt{37})}^{2}}{3^{2}})) - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Multiply $\frac{{(5 - \sqrt{37})}^{2}}{3^{2}}$ by $1$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + 15 (\frac{{(5 - \sqrt{37})}^{2}}{3^{2}}) - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + 15 (\frac{{(5 - \sqrt{37})}^{2}}{9}) - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Cancel the common factor of $3$ .

Factor $3$ out of $15$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + 3 (5) (\frac{{(5 - \sqrt{37})}^{2}}{9}) - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Factor $3$ out of $9$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + 3 \cdot (5 (\frac{{(5 - \sqrt{37})}^{2}}{3 \cdot 3})) - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Cancel the common factor.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + 3 \cdot (5 (\frac{{(5 - \sqrt{37})}^{2}}{3 \cdot 3})) - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Rewrite the expression.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + 5 (\frac{{(5 - \sqrt{37})}^{2}}{3}) - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Combine $5$ and $\frac{{(5 - \sqrt{37})}^{2}}{3}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 {(5 - \sqrt{37})}^{2}}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Rewrite ${(5 - \sqrt{37})}^{2}$ as $(5 - \sqrt{37}) (5 - \sqrt{37})$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 ((5 - \sqrt{37}) (5 - \sqrt{37}))}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Expand $(5 - \sqrt{37}) (5 - \sqrt{37})$ using the FOIL Method.

Apply the distributive property.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (5 (5 - \sqrt{37}) - \sqrt{37} (5 - \sqrt{37}))}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Apply the distributive property.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (5 \cdot 5 + 5 (- \sqrt{37}) - \sqrt{37} (5 - \sqrt{37}))}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Apply the distributive property.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (5 \cdot 5 + 5 (- \sqrt{37}) - \sqrt{37} \cdot 5 - \sqrt{37} (- \sqrt{37}))}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 + 5 (- \sqrt{37}) - \sqrt{37} \cdot 5 - \sqrt{37} (- \sqrt{37}))}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Multiply $- 1$ by $5$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - \sqrt{37} \cdot 5 - \sqrt{37} (- \sqrt{37}))}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Multiply $5$ by $- 1$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} - \sqrt{37} (- \sqrt{37}))}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Multiply $- \sqrt{37} (- \sqrt{37})$ .

Multiply $- 1$ by $- 1$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + 1 \sqrt{37} \sqrt{37})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Multiply $\sqrt{37}$ by $1$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + \sqrt{37} \sqrt{37})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Raise $\sqrt{37}$ to the power of $1$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + \sqrt{37} \sqrt{37})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Raise $\sqrt{37}$ to the power of $1$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + \sqrt{37} \sqrt{37})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + {\sqrt{37}}^{1 + 1})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Add $1$ and $1$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + {\sqrt{37}}^{2})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Rewrite ${\sqrt{37}}^{2}$ as $37$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{37}$ as $37^{\frac{1}{2}}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + {(37^{\frac{1}{2}})}^{2})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + 37^{\frac{1}{2} \cdot 2})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + 37^{\frac{2}{2}})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + 37^{\frac{2}{2}})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Rewrite the expression.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + 37)}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Evaluate the exponent.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (25 - 5 \sqrt{37} - 5 \sqrt{37} + 37)}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Add $25$ and $37$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 5 \sqrt{37} - 5 \sqrt{37})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Subtract $5 \sqrt{37}$ from $- 5 \sqrt{37}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37})}{3} - 12 (- \frac{5 - \sqrt{37}}{3}) - 60$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{5 - \sqrt{37}}{3}$ into the numerator.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37})}{3} - 12 \frac{- (5 - \sqrt{37})}{3} - 60$

Factor $3$ out of $- 12$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37})}{3} + 3 (- 4) (\frac{- (5 - \sqrt{37})}{3}) - 60$

Cancel the common factor.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37})}{3} + 3 \cdot (- 4 \frac{- (5 - \sqrt{37})}{3}) - 60$

Rewrite the expression.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37})}{3} - 4 (- (5 - \sqrt{37})) - 60$

Multiply $- 1$ by $- 4$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37})}{3} + 4 (5 - \sqrt{37}) - 60$

Apply the distributive property.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37})}{3} + 4 \cdot 5 + 4 (- \sqrt{37}) - 60$

Multiply $4$ by $5$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37})}{3} + 20 + 4 (- \sqrt{37}) - 60$

Multiply $- 1$ by $4$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37})}{3} + 20 - 4 \sqrt{37} - 60$

To write $\frac{5 (62 - 10 \sqrt{37})}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37})}{3} \cdot \frac{3}{3} + 20 - 4 \sqrt{37} - 60$

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5 (62 - 10 \sqrt{37})}{3}$ by $\frac{3}{3}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37}) \cdot 3}{3 \cdot 3} + 20 - 4 \sqrt{37} - 60$

Multiply $3$ by $3$ .

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{680 - 112 \sqrt{37}}{9} + \frac{5 (62 - 10 \sqrt{37}) \cdot 3}{9} + 20 - 4 \sqrt{37} - 60$

Combine the numerators over the common denominator.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- (680 - 112 \sqrt{37}) + 5 (62 - 10 \sqrt{37}) \cdot 3}{9} + 20 - 4 \sqrt{37} - 60$

Simplify the numerator.

Apply the distributive property.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 1 \cdot 680 - (- 112 \sqrt{37}) + 5 (62 - 10 \sqrt{37}) \cdot 3}{9} + 20 - 4 \sqrt{37} - 60$

Multiply $- 1$ by $680$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 680 - (- 112 \sqrt{37}) + 5 (62 - 10 \sqrt{37}) \cdot 3}{9} + 20 - 4 \sqrt{37} - 60$

Multiply $- 112$ by $- 1$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 680 + 112 \sqrt{37} + 5 (62 - 10 \sqrt{37}) \cdot 3}{9} + 20 - 4 \sqrt{37} - 60$

Apply the distributive property.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 680 + 112 \sqrt{37} + (5 \cdot 62 + 5 (- 10 \sqrt{37})) \cdot 3}{9} + 20 - 4 \sqrt{37} - 60$

Multiply $5$ by $62$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 680 + 112 \sqrt{37} + (310 + 5 (- 10 \sqrt{37})) \cdot 3}{9} + 20 - 4 \sqrt{37} - 60$

Multiply $- 10$ by $5$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 680 + 112 \sqrt{37} + (310 - 50 \sqrt{37}) \cdot 3}{9} + 20 - 4 \sqrt{37} - 60$

Apply the distributive property.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 680 + 112 \sqrt{37} + 310 \cdot 3 - 50 \sqrt{37} \cdot 3}{9} + 20 - 4 \sqrt{37} - 60$

Multiply $310$ by $3$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 680 + 112 \sqrt{37} + 930 - 50 \sqrt{37} \cdot 3}{9} + 20 - 4 \sqrt{37} - 60$

Multiply $3$ by $- 50$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 680 + 112 \sqrt{37} + 930 - 150 \sqrt{37}}{9} + 20 - 4 \sqrt{37} - 60$

Add $- 680$ and $930$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{250 + 112 \sqrt{37} - 150 \sqrt{37}}{9} + 20 - 4 \sqrt{37} - 60$

Subtract $150 \sqrt{37}$ from $112 \sqrt{37}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{250 - 38 \sqrt{37}}{9} + 20 - 4 \sqrt{37} - 60$

To write $20$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{250 - 38 \sqrt{37}}{9} + 20 \cdot \frac{9}{9} - 4 \sqrt{37} - 60$

Combine $20$ and $\frac{9}{9}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{250 - 38 \sqrt{37}}{9} + \frac{20 \cdot 9}{9} - 4 \sqrt{37} - 60$

Simplify the expression.

Combine the numerators over the common denominator.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{250 - 38 \sqrt{37} + 20 \cdot 9}{9} - 4 \sqrt{37} - 60$

Multiply $20$ by $9$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{250 - 38 \sqrt{37} + 180}{9} - 4 \sqrt{37} - 60$

Add $250$ and $180$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{430 - 38 \sqrt{37}}{9} - 4 \sqrt{37} - 60$

To write $- 4 \sqrt{37}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{430 - 38 \sqrt{37}}{9} - 4 \sqrt{37} \cdot \frac{9}{9} - 60$

Combine fractions.

Combine $- 4 \sqrt{37}$ and $\frac{9}{9}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{430 - 38 \sqrt{37}}{9} + \frac{- 4 \sqrt{37} \cdot 9}{9} - 60$

Combine the numerators over the common denominator.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{430 - 38 \sqrt{37} - 4 \sqrt{37} \cdot 9}{9} - 60$

Simplify the numerator.

Multiply $9$ by $- 4$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{430 - 38 \sqrt{37} - 36 \sqrt{37}}{9} - 60$

Subtract $36 \sqrt{37}$ from $- 38 \sqrt{37}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{430 - 74 \sqrt{37}}{9} - 60$

To write $- 60$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{430 - 74 \sqrt{37}}{9} - 60 \cdot \frac{9}{9}$

Combine fractions.

Combine $- 60$ and $\frac{9}{9}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{430 - 74 \sqrt{37}}{9} + \frac{- 60 \cdot 9}{9}$

Combine the numerators over the common denominator.

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{430 - 74 \sqrt{37} - 60 \cdot 9}{9}$

Simplify the numerator.

Multiply $- 60$ by $9$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{430 - 74 \sqrt{37} - 540}{9}$

Subtract $540$ from $430$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 110 - 74 \sqrt{37}}{9}$

Simplify with factoring out.

Rewrite $- 110$ as $- 1 (110)$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 1 \cdot 110 - 74 \sqrt{37}}{9}$

Factor $- 1$ out of $- 74 \sqrt{37}$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 1 \cdot 110 - (74 \sqrt{37})}{9}$

Factor $- 1$ out of $- 1 (110) - (74 \sqrt{37})$ .

$f (- \frac{5 - \sqrt{37}}{3}) = \frac{- 1 (110 + 74 \sqrt{37})}{9}$

Move the negative in front of the fraction.

$f (- \frac{5 - \sqrt{37}}{3}) = - \frac{110 + 74 \sqrt{37}}{9}$

The final answer is $- \frac{110 + 74 \sqrt{37}}{9}$ .

$y = - \frac{110 + 74 \sqrt{37}}{9}$

Step 13

Evaluate the second derivative at $x = - \frac{5 + \sqrt{37}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$18 (- \frac{5 + \sqrt{37}}{3}) + 30$

Step 14

Evaluate the second derivative.

Simplify each term.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{5 + \sqrt{37}}{3}$ into the numerator.

$18 \frac{- (5 + \sqrt{37})}{3} + 30$

Factor $3$ out of $18$ .

$3 (6) \frac{- (5 + \sqrt{37})}{3} + 30$

Cancel the common factor.

$3 \cdot 6 \frac{- (5 + \sqrt{37})}{3} + 30$

Rewrite the expression.

$6 (- (5 + \sqrt{37})) + 30$

Multiply $- 1$ by $6$ .

$- 6 (5 + \sqrt{37}) + 30$

Apply the distributive property.

$- 6 \cdot 5 - 6 \sqrt{37} + 30$

Multiply $- 6$ by $5$ .

$- 30 - 6 \sqrt{37} + 30$

Simplify by adding numbers.

Add $- 30$ and $30$ .

$0 - 6 \sqrt{37}$

Subtract $6 \sqrt{37}$ from $0$ .

$- 6 \sqrt{37}$

Step 15

$x = - \frac{5 + \sqrt{37}}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{5 + \sqrt{37}}{3}$ is a local maximum

Step 16

Find the y-value when $x = - \frac{5 + \sqrt{37}}{3}$ .

Replace the variable $x$ with $- \frac{5 + \sqrt{37}}{3}$ in the expression.

$f (- \frac{5 + \sqrt{37}}{3}) = 3 {(- \frac{5 + \sqrt{37}}{3})}^{3} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

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 $- \frac{5 + \sqrt{37}}{3}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = 3 ({(- 1)}^{3} {(\frac{5 + \sqrt{37}}{3})}^{3}) + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Apply the product rule to $\frac{5 + \sqrt{37}}{3}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = 3 ({(- 1)}^{3} (\frac{{(5 + \sqrt{37})}^{3}}{3^{3}})) + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 + \sqrt{37}}{3}) = 3 (- \frac{{(5 + \sqrt{37})}^{3}}{3^{3}}) + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 + \sqrt{37}}{3}) = 3 (- \frac{{(5 + \sqrt{37})}^{3}}{27}) + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{{(5 + \sqrt{37})}^{3}}{27}$ into the numerator.

$f (- \frac{5 + \sqrt{37}}{3}) = 3 (\frac{- {(5 + \sqrt{37})}^{3}}{27}) + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Factor $3$ out of $27$ .

$f (- \frac{5 + \sqrt{37}}{3}) = 3 (\frac{- {(5 + \sqrt{37})}^{3}}{3 (9)}) + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Cancel the common factor.

$f (- \frac{5 + \sqrt{37}}{3}) = 3 (\frac{- {(5 + \sqrt{37})}^{3}}{3 \cdot 9}) + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Rewrite the expression.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- {(5 + \sqrt{37})}^{3}}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Use the Binomial Theorem.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (5^{3} + 3 \cdot (5^{2} \sqrt{37}) + 3 \cdot (5 {\sqrt{37}}^{2}) + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Simplify each term.

Raise $5$ to the power of $3$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 3 \cdot (5^{2} \sqrt{37}) + 3 \cdot (5 {\sqrt{37}}^{2}) + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Raise $5$ to the power of $2$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 3 \cdot (25 \sqrt{37}) + 3 \cdot (5 {\sqrt{37}}^{2}) + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Multiply $3$ by $25$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 3 \cdot (5 {\sqrt{37}}^{2}) + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Multiply $3$ by $5$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 15 {\sqrt{37}}^{2} + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Rewrite ${\sqrt{37}}^{2}$ as $37$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{37}$ as $37^{\frac{1}{2}}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 15 {(37^{\frac{1}{2}})}^{2} + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 15 \cdot 37^{\frac{1}{2} \cdot 2} + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 15 \cdot 37^{\frac{2}{2}} + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 15 \cdot 37^{\frac{2}{2}} + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Rewrite the expression.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 15 \cdot 37 + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Evaluate the exponent.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 15 \cdot 37 + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Multiply $15$ by $37$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 555 + {\sqrt{37}}^{3})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Rewrite ${\sqrt{37}}^{3}$ as ${(37^{3})}^{\frac{1}{2}}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 555 + \sqrt{37^{3}})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Raise $37$ to the power of $3$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 555 + \sqrt{50653})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Rewrite $50653$ as $37^{2} \cdot 37$ .

Factor $1369$ out of $50653$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 555 + \sqrt{1369 (37)})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Rewrite $1369$ as $37^{2}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 555 + \sqrt{37^{2} \cdot 37})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Pull terms out from under the radical.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (125 + 75 \sqrt{37} + 555 + 37 \sqrt{37})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Add $125$ and $555$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (680 + 75 \sqrt{37} + 37 \sqrt{37})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Add $75 \sqrt{37}$ and $37 \sqrt{37}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (680 + 112 \sqrt{37})}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Move the negative in front of the fraction.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + 15 {(- \frac{5 + \sqrt{37}}{3})}^{2} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

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

Apply the product rule to $- \frac{5 + \sqrt{37}}{3}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + 15 ({(- 1)}^{2} {(\frac{5 + \sqrt{37}}{3})}^{2}) - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Apply the product rule to $\frac{5 + \sqrt{37}}{3}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + 15 ({(- 1)}^{2} (\frac{{(5 + \sqrt{37})}^{2}}{3^{2}})) - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + 15 (1 (\frac{{(5 + \sqrt{37})}^{2}}{3^{2}})) - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Multiply $\frac{{(5 + \sqrt{37})}^{2}}{3^{2}}$ by $1$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + 15 (\frac{{(5 + \sqrt{37})}^{2}}{3^{2}}) - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + 15 (\frac{{(5 + \sqrt{37})}^{2}}{9}) - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Cancel the common factor of $3$ .

Factor $3$ out of $15$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + 3 (5) (\frac{{(5 + \sqrt{37})}^{2}}{9}) - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Factor $3$ out of $9$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + 3 \cdot (5 (\frac{{(5 + \sqrt{37})}^{2}}{3 \cdot 3})) - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Cancel the common factor.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + 3 \cdot (5 (\frac{{(5 + \sqrt{37})}^{2}}{3 \cdot 3})) - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Rewrite the expression.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + 5 (\frac{{(5 + \sqrt{37})}^{2}}{3}) - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Combine $5$ and $\frac{{(5 + \sqrt{37})}^{2}}{3}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 {(5 + \sqrt{37})}^{2}}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Rewrite ${(5 + \sqrt{37})}^{2}$ as $(5 + \sqrt{37}) (5 + \sqrt{37})$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 ((5 + \sqrt{37}) (5 + \sqrt{37}))}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Expand $(5 + \sqrt{37}) (5 + \sqrt{37})$ using the FOIL Method.

Apply the distributive property.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (5 (5 + \sqrt{37}) + \sqrt{37} (5 + \sqrt{37}))}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Apply the distributive property.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (5 \cdot 5 + 5 \sqrt{37} + \sqrt{37} (5 + \sqrt{37}))}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Apply the distributive property.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (5 \cdot 5 + 5 \sqrt{37} + \sqrt{37} \cdot 5 + \sqrt{37} \sqrt{37})}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (25 + 5 \sqrt{37} + \sqrt{37} \cdot 5 + \sqrt{37} \sqrt{37})}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Move $5$ to the left of $\sqrt{37}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (25 + 5 \sqrt{37} + 5 \cdot \sqrt{37} + \sqrt{37} \sqrt{37})}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Combine using the product rule for radicals.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (25 + 5 \sqrt{37} + 5 \sqrt{37} + \sqrt{37 \cdot 37})}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Multiply $37$ by $37$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (25 + 5 \sqrt{37} + 5 \sqrt{37} + \sqrt{1369})}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Rewrite $1369$ as $37^{2}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (25 + 5 \sqrt{37} + 5 \sqrt{37} + \sqrt{37^{2}})}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

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

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (25 + 5 \sqrt{37} + 5 \sqrt{37} + 37)}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Add $25$ and $37$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 5 \sqrt{37} + 5 \sqrt{37})}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Add $5 \sqrt{37}$ and $5 \sqrt{37}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 10 \sqrt{37})}{3} - 12 (- \frac{5 + \sqrt{37}}{3}) - 60$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{5 + \sqrt{37}}{3}$ into the numerator.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 10 \sqrt{37})}{3} - 12 \frac{- (5 + \sqrt{37})}{3} - 60$

Factor $3$ out of $- 12$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 10 \sqrt{37})}{3} + 3 (- 4) (\frac{- (5 + \sqrt{37})}{3}) - 60$

Cancel the common factor.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 10 \sqrt{37})}{3} + 3 \cdot (- 4 \frac{- (5 + \sqrt{37})}{3}) - 60$

Rewrite the expression.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 10 \sqrt{37})}{3} - 4 (- (5 + \sqrt{37})) - 60$

Multiply $- 1$ by $- 4$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 10 \sqrt{37})}{3} + 4 (5 + \sqrt{37}) - 60$

Apply the distributive property.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 10 \sqrt{37})}{3} + 4 \cdot 5 + 4 \sqrt{37} - 60$

Multiply $4$ by $5$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 10 \sqrt{37})}{3} + 20 + 4 \sqrt{37} - 60$

To write $\frac{5 (62 + 10 \sqrt{37})}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 10 \sqrt{37})}{3} \cdot \frac{3}{3} + 20 + 4 \sqrt{37} - 60$

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5 (62 + 10 \sqrt{37})}{3}$ by $\frac{3}{3}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 10 \sqrt{37}) \cdot 3}{3 \cdot 3} + 20 + 4 \sqrt{37} - 60$

Multiply $3$ by $3$ .

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{680 + 112 \sqrt{37}}{9} + \frac{5 (62 + 10 \sqrt{37}) \cdot 3}{9} + 20 + 4 \sqrt{37} - 60$

Combine the numerators over the common denominator.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- (680 + 112 \sqrt{37}) + 5 (62 + 10 \sqrt{37}) \cdot 3}{9} + 20 + 4 \sqrt{37} - 60$

Simplify the numerator.

Apply the distributive property.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 1 \cdot 680 - (112 \sqrt{37}) + 5 (62 + 10 \sqrt{37}) \cdot 3}{9} + 20 + 4 \sqrt{37} - 60$

Multiply $- 1$ by $680$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 680 - (112 \sqrt{37}) + 5 (62 + 10 \sqrt{37}) \cdot 3}{9} + 20 + 4 \sqrt{37} - 60$

Multiply $112$ by $- 1$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 680 - 112 \sqrt{37} + 5 (62 + 10 \sqrt{37}) \cdot 3}{9} + 20 + 4 \sqrt{37} - 60$

Apply the distributive property.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 680 - 112 \sqrt{37} + (5 \cdot 62 + 5 (10 \sqrt{37})) \cdot 3}{9} + 20 + 4 \sqrt{37} - 60$

Multiply $5$ by $62$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 680 - 112 \sqrt{37} + (310 + 5 (10 \sqrt{37})) \cdot 3}{9} + 20 + 4 \sqrt{37} - 60$

Multiply $10$ by $5$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 680 - 112 \sqrt{37} + (310 + 50 \sqrt{37}) \cdot 3}{9} + 20 + 4 \sqrt{37} - 60$

Apply the distributive property.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 680 - 112 \sqrt{37} + 310 \cdot 3 + 50 \sqrt{37} \cdot 3}{9} + 20 + 4 \sqrt{37} - 60$

Multiply $310$ by $3$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 680 - 112 \sqrt{37} + 930 + 50 \sqrt{37} \cdot 3}{9} + 20 + 4 \sqrt{37} - 60$

Multiply $3$ by $50$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 680 - 112 \sqrt{37} + 930 + 150 \sqrt{37}}{9} + 20 + 4 \sqrt{37} - 60$

Add $- 680$ and $930$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{250 - 112 \sqrt{37} + 150 \sqrt{37}}{9} + 20 + 4 \sqrt{37} - 60$

Add $- 112 \sqrt{37}$ and $150 \sqrt{37}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{250 + 38 \sqrt{37}}{9} + 20 + 4 \sqrt{37} - 60$

To write $20$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{250 + 38 \sqrt{37}}{9} + 20 \cdot \frac{9}{9} + 4 \sqrt{37} - 60$

Combine $20$ and $\frac{9}{9}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{250 + 38 \sqrt{37}}{9} + \frac{20 \cdot 9}{9} + 4 \sqrt{37} - 60$

Simplify the expression.

Combine the numerators over the common denominator.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{250 + 38 \sqrt{37} + 20 \cdot 9}{9} + 4 \sqrt{37} - 60$

Multiply $20$ by $9$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{250 + 38 \sqrt{37} + 180}{9} + 4 \sqrt{37} - 60$

Add $250$ and $180$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{430 + 38 \sqrt{37}}{9} + 4 \sqrt{37} - 60$

To write $4 \sqrt{37}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{430 + 38 \sqrt{37}}{9} + 4 \sqrt{37} \cdot \frac{9}{9} - 60$

Combine fractions.

Combine $4 \sqrt{37}$ and $\frac{9}{9}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{430 + 38 \sqrt{37}}{9} + \frac{4 \sqrt{37} \cdot 9}{9} - 60$

Combine the numerators over the common denominator.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{430 + 38 \sqrt{37} + 4 \sqrt{37} \cdot 9}{9} - 60$

Simplify the numerator.

Multiply $9$ by $4$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{430 + 38 \sqrt{37} + 36 \sqrt{37}}{9} - 60$

Add $38 \sqrt{37}$ and $36 \sqrt{37}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{430 + 74 \sqrt{37}}{9} - 60$

To write $- 60$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{430 + 74 \sqrt{37}}{9} - 60 \cdot \frac{9}{9}$

Combine fractions.

Combine $- 60$ and $\frac{9}{9}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{430 + 74 \sqrt{37}}{9} + \frac{- 60 \cdot 9}{9}$

Combine the numerators over the common denominator.

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{430 + 74 \sqrt{37} - 60 \cdot 9}{9}$

Simplify the numerator.

Multiply $- 60$ by $9$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{430 + 74 \sqrt{37} - 540}{9}$

Subtract $540$ from $430$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 110 + 74 \sqrt{37}}{9}$

Simplify with factoring out.

Rewrite $- 110$ as $- 1 (110)$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 1 \cdot 110 + 74 \sqrt{37}}{9}$

Factor $- 1$ out of $74 \sqrt{37}$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 1 \cdot 110 - (- 74 \sqrt{37})}{9}$

Factor $- 1$ out of $- 1 (110) - (- 74 \sqrt{37})$ .

$f (- \frac{5 + \sqrt{37}}{3}) = \frac{- 1 (110 - 74 \sqrt{37})}{9}$

Move the negative in front of the fraction.

$f (- \frac{5 + \sqrt{37}}{3}) = - \frac{110 - 74 \sqrt{37}}{9}$

The final answer is $- \frac{110 - 74 \sqrt{37}}{9}$ .

$y = - \frac{110 - 74 \sqrt{37}}{9}$

Step 17

These are the local extrema for $f (x) = 3 x^{3} + 15 x^{2} - 12 x - 60$ .

$(- \frac{5 - \sqrt{37}}{3}, - \frac{110 + 74 \sqrt{37}}{9})$ is a local minima

$(- \frac{5 + \sqrt{37}}{3}, - \frac{110 - 74 \sqrt{37}}{9})$ is a local maxima

Step 18

Find the Local Maxima and Minima y=3x^3+15x^2-12x-60

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