Find the Maximum/Minimum Value x(16-2x)(14-2x)

$x (16 - 2 x) (14 - 2 x)$

Step 1

Find the first derivative of the function.

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x (16 - 2 x)$ and $g (x) = 14 - 2 x$ .

$x (16 - 2 x) \frac{d}{d x} [14 - 2 x] + (14 - 2 x) \frac{d}{d x} [x (16 - 2 x)]$

Differentiate.

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

$x (16 - 2 x) (\frac{d}{d x} [14] + \frac{d}{d x} [- 2 x]) + (14 - 2 x) \frac{d}{d x} [x (16 - 2 x)]$

Since $14$ is constant with respect to $x$ , the derivative of $14$ with respect to $x$ is $0$ .

$x (16 - 2 x) (0 + \frac{d}{d x} [- 2 x]) + (14 - 2 x) \frac{d}{d x} [x (16 - 2 x)]$

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$x (16 - 2 x) \frac{d}{d x} [- 2 x] + (14 - 2 x) \frac{d}{d x} [x (16 - 2 x)]$

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

$x (16 - 2 x) (- 2 \frac{d}{d x} [x]) + (14 - 2 x) \frac{d}{d x} [x (16 - 2 x)]$

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

$x (16 - 2 x) (- 2 \cdot 1) + (14 - 2 x) \frac{d}{d x} [x (16 - 2 x)]$

Simplify the expression.

Multiply $- 2$ by $1$ .

$x (16 - 2 x) \cdot - 2 + (14 - 2 x) \frac{d}{d x} [x (16 - 2 x)]$

Move $- 2$ to the left of $x (16 - 2 x)$ .

$- 2 \cdot (x (16 - 2 x)) + (14 - 2 x) \frac{d}{d x} [x (16 - 2 x)]$

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 16 - 2 x$ .

$- 2 x (16 - 2 x) + (14 - 2 x) (x \frac{d}{d x} [16 - 2 x] + (16 - 2 x) \frac{d}{d x} [x])$

Differentiate.

By the Sum Rule, the derivative of $16 - 2 x$ with respect to $x$ is $\frac{d}{d x} [16] + \frac{d}{d x} [- 2 x]$ .

$- 2 x (16 - 2 x) + (14 - 2 x) (x (\frac{d}{d x} [16] + \frac{d}{d x} [- 2 x]) + (16 - 2 x) \frac{d}{d x} [x])$

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$- 2 x (16 - 2 x) + (14 - 2 x) (x (0 + \frac{d}{d x} [- 2 x]) + (16 - 2 x) \frac{d}{d x} [x])$

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$- 2 x (16 - 2 x) + (14 - 2 x) (x \frac{d}{d x} [- 2 x] + (16 - 2 x) \frac{d}{d x} [x])$

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

$- 2 x (16 - 2 x) + (14 - 2 x) (x (- 2 \frac{d}{d x} [x]) + (16 - 2 x) \frac{d}{d x} [x])$

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

$- 2 x (16 - 2 x) + (14 - 2 x) (x (- 2 \cdot 1) + (16 - 2 x) \frac{d}{d x} [x])$

Simplify the expression.

Multiply $- 2$ by $1$ .

$- 2 x (16 - 2 x) + (14 - 2 x) (x \cdot - 2 + (16 - 2 x) \frac{d}{d x} [x])$

Move $- 2$ to the left of $x$ .

$- 2 x (16 - 2 x) + (14 - 2 x) (- 2 \cdot x + (16 - 2 x) \frac{d}{d x} [x])$

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

$- 2 x (16 - 2 x) + (14 - 2 x) (- 2 x + (16 - 2 x) \cdot 1)$

Simplify by adding terms.

Multiply $16 - 2 x$ by $1$ .

$- 2 x (16 - 2 x) + (14 - 2 x) (- 2 x + 16 - 2 x)$

Subtract $2 x$ from $- 2 x$ .

$- 2 x (16 - 2 x) + (14 - 2 x) (- 4 x + 16)$

Simplify.

Apply the distributive property.

$- 2 x \cdot 16 - 2 x (- 2 x) + (14 - 2 x) (- 4 x + 16)$

Apply the distributive property.

$- 2 x \cdot 16 - 2 x (- 2 x) + (14 - 2 x) (- 4 x) + (14 - 2 x) \cdot 16$

Apply the distributive property.

$- 2 x \cdot 16 - 2 x (- 2 x) + 14 (- 4 x) - 2 x (- 4 x) + (14 - 2 x) \cdot 16$

Apply the distributive property.

$- 2 x \cdot 16 - 2 x (- 2 x) + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Combine terms.

Multiply $16$ by $- 2$ .

$- 32 x - 2 x (- 2 x) + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Multiply $- 2$ by $- 2$ .

$- 32 x + 4 x \cdot x + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Raise $x$ to the power of $1$ .

$- 32 x + 4 (x^{1} x) + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Raise $x$ to the power of $1$ .

$- 32 x + 4 (x^{1} x^{1}) + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

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

$- 32 x + 4 x^{1 + 1} + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Add $1$ and $1$ .

$- 32 x + 4 x^{2} + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Multiply $- 4$ by $14$ .

$- 32 x + 4 x^{2} - 56 x - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Multiply $- 4$ by $- 2$ .

$- 32 x + 4 x^{2} - 56 x + 8 x \cdot x + 14 \cdot 16 - 2 x \cdot 16$

Raise $x$ to the power of $1$ .

$- 32 x + 4 x^{2} - 56 x + 8 (x^{1} x) + 14 \cdot 16 - 2 x \cdot 16$

Raise $x$ to the power of $1$ .

$- 32 x + 4 x^{2} - 56 x + 8 (x^{1} x^{1}) + 14 \cdot 16 - 2 x \cdot 16$

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

$- 32 x + 4 x^{2} - 56 x + 8 x^{1 + 1} + 14 \cdot 16 - 2 x \cdot 16$

Add $1$ and $1$ .

$- 32 x + 4 x^{2} - 56 x + 8 x^{2} + 14 \cdot 16 - 2 x \cdot 16$

Multiply $14$ by $16$ .

$- 32 x + 4 x^{2} - 56 x + 8 x^{2} + 224 - 2 x \cdot 16$

Multiply $16$ by $- 2$ .

$- 32 x + 4 x^{2} - 56 x + 8 x^{2} + 224 - 32 x$

Subtract $32 x$ from $- 56 x$ .

$- 32 x + 4 x^{2} - 88 x + 8 x^{2} + 224$

Subtract $88 x$ from $- 32 x$ .

$4 x^{2} - 120 x + 8 x^{2} + 224$

Add $4 x^{2}$ and $8 x^{2}$ .

$12 x^{2} - 120 x + 224$

Step 2

Find the second derivative of the function.

By the Sum Rule, the derivative of $12 x^{2} - 120 x + 224$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 120 x] + \frac{d}{d x} [224]$ .

$f'' (x) = \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (- 120 x) + \frac{d}{d x} (224)$

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 120 x) + \frac{d}{d x} (224)$

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

$f'' (x) = 12 (2 x) + \frac{d}{d x} (- 120 x) + \frac{d}{d x} (224)$

Multiply $2$ by $12$ .

$f'' (x) = 24 x + \frac{d}{d x} (- 120 x) + \frac{d}{d x} (224)$

Evaluate $\frac{d}{d x} [- 120 x]$ .

Since $- 120$ is constant with respect to $x$ , the derivative of $- 120 x$ with respect to $x$ is $- 120 \frac{d}{d x} [x]$ .

$f'' (x) = 24 x - 120 \frac{d}{d x} x + \frac{d}{d x} (224)$

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

$f'' (x) = 24 x - 120 \cdot 1 + \frac{d}{d x} (224)$

Multiply $- 120$ by $1$ .

$f'' (x) = 24 x - 120 + \frac{d}{d x} (224)$

Differentiate using the Constant Rule.

Since $224$ is constant with respect to $x$ , the derivative of $224$ with respect to $x$ is $0$ .

$f'' (x) = 24 x - 120 + 0$

Add $24 x - 120$ and $0$ .

$f'' (x) = 24 x - 120$

Step 3

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

$12 x^{2} - 120 x + 224 = 0$

Step 4

Find the first derivative.

$f' (x) = x (16 - 2 x) \frac{d}{d x} (14 - 2 x) + (14 - 2 x) \frac{d}{d x} (x (16 - 2 x))$

Differentiate.

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

$f' (x) = x (16 - 2 x) (\frac{d}{d x} (14) + \frac{d}{d x} (- 2 x)) + (14 - 2 x) \frac{d}{d x} (x (16 - 2 x))$

Since $14$ is constant with respect to $x$ , the derivative of $14$ with respect to $x$ is $0$ .

$f' (x) = x (16 - 2 x) (0 + \frac{d}{d x} (- 2 x)) + (14 - 2 x) \frac{d}{d x} (x (16 - 2 x))$

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$f' (x) = x (16 - 2 x) \frac{d}{d x} (- 2 x) + (14 - 2 x) \frac{d}{d x} (x (16 - 2 x))$

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

$f' (x) = x (16 - 2 x) (- 2 \frac{d}{d x} x) + (14 - 2 x) \frac{d}{d x} (x (16 - 2 x))$

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

$f' (x) = x (16 - 2 x) (- 2 \cdot 1) + (14 - 2 x) \frac{d}{d x} (x (16 - 2 x))$

Simplify the expression.

Multiply $- 2$ by $1$ .

$f' (x) = x (16 - 2 x) \cdot - 2 + (14 - 2 x) \frac{d}{d x} (x (16 - 2 x))$

Move $- 2$ to the left of $x (16 - 2 x)$ .

$f' (x) = - 2 \cdot (x (16 - 2 x)) + (14 - 2 x) \frac{d}{d x} (x (16 - 2 x))$

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 16 - 2 x$ .

$f' (x) = - 2 x (16 - 2 x) + (14 - 2 x) (x \frac{d}{d x} (16 - 2 x) + (16 - 2 x) \frac{d}{d x} (x))$

Differentiate.

By the Sum Rule, the derivative of $16 - 2 x$ with respect to $x$ is $\frac{d}{d x} [16] + \frac{d}{d x} [- 2 x]$ .

$f' (x) = - 2 x (16 - 2 x) + (14 - 2 x) (x (\frac{d}{d x} (16) + \frac{d}{d x} (- 2 x)) + (16 - 2 x) \frac{d}{d x} (x))$

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$f' (x) = - 2 x (16 - 2 x) + (14 - 2 x) (x (0 + \frac{d}{d x} (- 2 x)) + (16 - 2 x) \frac{d}{d x} (x))$

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$f' (x) = - 2 x (16 - 2 x) + (14 - 2 x) (x \frac{d}{d x} (- 2 x) + (16 - 2 x) \frac{d}{d x} (x))$

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

$f' (x) = - 2 x (16 - 2 x) + (14 - 2 x) (x (- 2 \frac{d}{d x} x) + (16 - 2 x) \frac{d}{d x} (x))$

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

$f' (x) = - 2 x (16 - 2 x) + (14 - 2 x) (x (- 2 \cdot 1) + (16 - 2 x) \frac{d}{d x} (x))$

Simplify the expression.

Multiply $- 2$ by $1$ .

$f' (x) = - 2 x (16 - 2 x) + (14 - 2 x) (x \cdot - 2 + (16 - 2 x) \frac{d}{d x} (x))$

Move $- 2$ to the left of $x$ .

$f' (x) = - 2 x (16 - 2 x) + (14 - 2 x) (- 2 \cdot x + (16 - 2 x) \frac{d}{d x} (x))$

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

$f' (x) = - 2 x (16 - 2 x) + (14 - 2 x) (- 2 x + (16 - 2 x) \cdot 1)$

Simplify by adding terms.

Multiply $16 - 2 x$ by $1$ .

$f' (x) = - 2 x (16 - 2 x) + (14 - 2 x) (- 2 x + 16 - 2 x)$

Subtract $2 x$ from $- 2 x$ .

$f' (x) = - 2 x (16 - 2 x) + (14 - 2 x) (- 4 x + 16)$

Simplify.

Apply the distributive property.

$f' (x) = - 2 x \cdot 16 - 2 x (- 2 x) + (14 - 2 x) (- 4 x + 16)$

Apply the distributive property.

$f' (x) = - 2 x \cdot 16 - 2 x (- 2 x) + (14 - 2 x) (- 4 x) + (14 - 2 x) \cdot 16$

Apply the distributive property.

$f' (x) = - 2 x \cdot 16 - 2 x (- 2 x) + 14 (- 4 x) - 2 x (- 4 x) + (14 - 2 x) \cdot 16$

Apply the distributive property.

$f' (x) = - 2 x \cdot 16 - 2 x (- 2 x) + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Combine terms.

Multiply $16$ by $- 2$ .

$f' (x) = - 32 x - 2 x (- 2 x) + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Multiply $- 2$ by $- 2$ .

$f' (x) = - 32 x + 4 x \cdot x + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Raise $x$ to the power of $1$ .

$f' (x) = - 32 x + 4 (x \cdot x) + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Raise $x$ to the power of $1$ .

$f' (x) = - 32 x + 4 (x \cdot x) + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

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

$f' (x) = - 32 x + 4 x^{1 + 1} + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Add $1$ and $1$ .

$f' (x) = - 32 x + 4 x^{2} + 14 (- 4 x) - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Multiply $- 4$ by $14$ .

$f' (x) = - 32 x + 4 x^{2} - 56 x - 2 x (- 4 x) + 14 \cdot 16 - 2 x \cdot 16$

Multiply $- 4$ by $- 2$ .

$f' (x) = - 32 x + 4 x^{2} - 56 x + 8 x \cdot x + 14 \cdot 16 - 2 x \cdot 16$

Raise $x$ to the power of $1$ .

$f' (x) = - 32 x + 4 x^{2} - 56 x + 8 (x \cdot x) + 14 \cdot 16 - 2 x \cdot 16$

Raise $x$ to the power of $1$ .

$f' (x) = - 32 x + 4 x^{2} - 56 x + 8 (x \cdot x) + 14 \cdot 16 - 2 x \cdot 16$

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

$f' (x) = - 32 x + 4 x^{2} - 56 x + 8 x^{1 + 1} + 14 \cdot 16 - 2 x \cdot 16$

Add $1$ and $1$ .

$f' (x) = - 32 x + 4 x^{2} - 56 x + 8 x^{2} + 14 \cdot 16 - 2 x \cdot 16$

Multiply $14$ by $16$ .

$f' (x) = - 32 x + 4 x^{2} - 56 x + 8 x^{2} + 224 - 2 x \cdot 16$

Multiply $16$ by $- 2$ .

$f' (x) = - 32 x + 4 x^{2} - 56 x + 8 x^{2} + 224 - 32 x$

Subtract $32 x$ from $- 56 x$ .

$f' (x) = - 32 x + 4 x^{2} - 88 x + 8 x^{2} + 224$

Subtract $88 x$ from $- 32 x$ .

$f' (x) = 4 x^{2} - 120 x + 8 x^{2} + 224$

Add $4 x^{2}$ and $8 x^{2}$ .

$f' (x) = 12 x^{2} - 120 x + 224$

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 120 x + 224$ .

$12 x^{2} - 120 x + 224$

Step 5

Set the first derivative equal to $0$ then solve the equation $12 x^{2} - 120 x + 224 = 0$ .

Set the first derivative equal to $0$ .

$12 x^{2} - 120 x + 224 = 0$

Factor $4$ out of $12 x^{2} - 120 x + 224$ .

Factor $4$ out of $12 x^{2}$ .

$4 (3 x^{2}) - 120 x + 224 = 0$

Factor $4$ out of $- 120 x$ .

$4 (3 x^{2}) + 4 (- 30 x) + 224 = 0$

Factor $4$ out of $224$ .

$4 (3 x^{2}) + 4 (- 30 x) + 4 (56) = 0$

Factor $4$ out of $4 (3 x^{2}) + 4 (- 30 x)$ .

$4 (3 x^{2} - 30 x) + 4 (56) = 0$

Factor $4$ out of $4 (3 x^{2} - 30 x) + 4 (56)$ .

$4 (3 x^{2} - 30 x + 56) = 0$

Divide each term in $4 (3 x^{2} - 30 x + 56) = 0$ by $4$ and simplify.

Divide each term in $4 (3 x^{2} - 30 x + 56) = 0$ by $4$ .

$\frac{4 (3 x^{2} - 30 x + 56)}{4} = \frac{0}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 (3 x^{2} - 30 x + 56)}{4} = \frac{0}{4}$

Divide $3 x^{2} - 30 x + 56$ by $1$ .

$3 x^{2} - 30 x + 56 = \frac{0}{4}$

Simplify the right side.

Divide $0$ by $4$ .

$3 x^{2} - 30 x + 56 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 3$ , $b = - 30$ , and $c = 56$ into the quadratic formula and solve for $x$ .

$\frac{30 \pm \sqrt{{(- 30)}^{2} - 4 \cdot (3 \cdot 56)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raise $- 30$ to the power of $2$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 3 \cdot 56}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 56$ .

Multiply $- 4$ by $3$ .

$x = \frac{30 \pm \sqrt{900 - 12 \cdot 56}}{2 \cdot 3}$

Multiply $- 12$ by $56$ .

$x = \frac{30 \pm \sqrt{900 - 672}}{2 \cdot 3}$

Subtract $672$ from $900$ .

$x = \frac{30 \pm \sqrt{228}}{2 \cdot 3}$

Rewrite $228$ as $2^{2} \cdot 57$ .

Factor $4$ out of $228$ .

$x = \frac{30 \pm \sqrt{4 (57)}}{2 \cdot 3}$

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

$x = \frac{30 \pm \sqrt{2^{2} \cdot 57}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{30 \pm 2 \sqrt{57}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{30 \pm 2 \sqrt{57}}{6}$

Simplify $\frac{30 \pm 2 \sqrt{57}}{6}$ .

$x = \frac{15 \pm \sqrt{57}}{3}$

The final answer is the combination of both solutions.

$x = \frac{15 + \sqrt{57}}{3}, \frac{15 - \sqrt{57}}{3}$

Step 6

Find the values where the derivative is undefined.

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

Step 7

Critical points to evaluate.

$x = \frac{15 + \sqrt{57}}{3}, \frac{15 - \sqrt{57}}{3}$

Step 8

Evaluate the second derivative at $x = \frac{15 + \sqrt{57}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$24 (\frac{15 + \sqrt{57}}{3}) - 120$

Step 9

Evaluate the second derivative.

Simplify each term.

Cancel the common factor of $3$ .

Factor $3$ out of $24$ .

$3 (8) \frac{15 + \sqrt{57}}{3} - 120$

Cancel the common factor.

$3 \cdot 8 \frac{15 + \sqrt{57}}{3} - 120$

Rewrite the expression.

$8 (15 + \sqrt{57}) - 120$

Apply the distributive property.

$8 \cdot 15 + 8 \sqrt{57} - 120$

Multiply $8$ by $15$ .

$120 + 8 \sqrt{57} - 120$

Simplify by subtracting numbers.

Subtract $120$ from $120$ .

$0 + 8 \sqrt{57}$

Add $0$ and $8 \sqrt{57}$ .

$8 \sqrt{57}$

Step 10

$x = \frac{15 + \sqrt{57}}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{15 + \sqrt{57}}{3}$ is a local minimum

Step 11

Find the y-value when $x = \frac{15 + \sqrt{57}}{3}$ .

Replace the variable $x$ with $\frac{15 + \sqrt{57}}{3}$ in the expression.

$f (\frac{15 + \sqrt{57}}{3}) = (\frac{15 + \sqrt{57}}{3}) (16 - 2 \frac{15 + \sqrt{57}}{3}) (14 - 2 \frac{15 + \sqrt{57}}{3})$

Simplify the result.

Simplify each term.

Combine $- 2$ and $\frac{15 + \sqrt{57}}{3}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 + \sqrt{57}}{3} \cdot ((16 + \frac{- 2 (15 + \sqrt{57})}{3}) (14 - 2 \frac{15 + \sqrt{57}}{3}))$

Move the negative in front of the fraction.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 + \sqrt{57}}{3} \cdot ((16 - \frac{2 (15 + \sqrt{57})}{3}) (14 - 2 \frac{15 + \sqrt{57}}{3}))$

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 + \sqrt{57}}{3} \cdot ((16 \cdot \frac{3}{3} - \frac{2 (15 + \sqrt{57})}{3}) (14 - 2 \frac{15 + \sqrt{57}}{3}))$

Combine fractions.

Combine $16$ and $\frac{3}{3}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 + \sqrt{57}}{3} \cdot ((\frac{16 \cdot 3}{3} - \frac{2 (15 + \sqrt{57})}{3}) (14 - 2 \frac{15 + \sqrt{57}}{3}))$

Combine the numerators over the common denominator.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 + \sqrt{57}}{3} \cdot \frac{16 \cdot 3 - 2 (15 + \sqrt{57})}{3} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Simplify the numerator.

Multiply $16$ by $3$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 + \sqrt{57}}{3} \cdot \frac{48 - 2 (15 + \sqrt{57})}{3} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Apply the distributive property.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 + \sqrt{57}}{3} \cdot \frac{48 - 2 \cdot 15 - 2 \sqrt{57}}{3} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Multiply $- 2$ by $15$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 + \sqrt{57}}{3} \cdot \frac{48 - 30 - 2 \sqrt{57}}{3} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Subtract $30$ from $48$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 + \sqrt{57}}{3} \cdot \frac{18 - 2 \sqrt{57}}{3} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Multiply $\frac{15 + \sqrt{57}}{3} \cdot \frac{18 - 2 \sqrt{57}}{3}$ .

Multiply $\frac{15 + \sqrt{57}}{3}$ by $\frac{18 - 2 \sqrt{57}}{3}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{(15 + \sqrt{57}) (18 - 2 \sqrt{57})}{3 \cdot 3} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Multiply $3$ by $3$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{(15 + \sqrt{57}) (18 - 2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Expand $(15 + \sqrt{57}) (18 - 2 \sqrt{57})$ using the FOIL Method.

Apply the distributive property.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 (18 - 2 \sqrt{57}) + \sqrt{57} (18 - 2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Apply the distributive property.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 \cdot 18 + 15 (- 2 \sqrt{57}) + \sqrt{57} (18 - 2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Apply the distributive property.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{15 \cdot 18 + 15 (- 2 \sqrt{57}) + \sqrt{57} \cdot 18 + \sqrt{57} (- 2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Simplify and combine like terms.

Simplify each term.

Multiply $15$ by $18$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 + 15 (- 2 \sqrt{57}) + \sqrt{57} \cdot 18 + \sqrt{57} (- 2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Multiply $- 2$ by $15$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + \sqrt{57} \cdot 18 + \sqrt{57} (- 2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Move $18$ to the left of $\sqrt{57}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \cdot \sqrt{57} + \sqrt{57} (- 2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Multiply $\sqrt{57} (- 2 \sqrt{57})$ .

Raise $\sqrt{57}$ to the power of $1$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \sqrt{57} - 2 (\sqrt{57} \sqrt{57})}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Raise $\sqrt{57}$ to the power of $1$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \sqrt{57} - 2 (\sqrt{57} \sqrt{57})}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

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

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \sqrt{57} - 2 {\sqrt{57}}^{1 + 1}}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Add $1$ and $1$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \sqrt{57} - 2 {\sqrt{57}}^{2}}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \sqrt{57} - 2 {(57^{\frac{1}{2}})}^{2}}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

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

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \sqrt{57} - 2 \cdot 57^{\frac{1}{2} \cdot 2}}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

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

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \sqrt{57} - 2 \cdot 57^{\frac{2}{2}}}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \sqrt{57} - 2 \cdot 57^{\frac{2}{2}}}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Rewrite the expression.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \sqrt{57} - 2 \cdot 57}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Evaluate the exponent.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \sqrt{57} - 2 \cdot 57}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Multiply $- 2$ by $57$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{270 - 30 \sqrt{57} + 18 \sqrt{57} - 114}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Subtract $114$ from $270$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{156 - 30 \sqrt{57} + 18 \sqrt{57}}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Add $- 30 \sqrt{57}$ and $18 \sqrt{57}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{156 - 12 \sqrt{57}}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Cancel the common factor of $156 - 12 \sqrt{57}$ and $9$ .

Factor $3$ out of $156$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{3 (52) - 12 \sqrt{57}}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Factor $3$ out of $- 12 \sqrt{57}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{3 (52) + 3 (- 4 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Factor $3$ out of $3 (52) + 3 (- 4 \sqrt{57})$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{3 (52 - 4 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Cancel the common factors.

Factor $3$ out of $9$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{3 (52 - 4 \sqrt{57})}{3 \cdot 3} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Cancel the common factor.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{3 (52 - 4 \sqrt{57})}{3 \cdot 3} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Rewrite the expression.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 - 4 \sqrt{57}}{3} \cdot (14 - 2 \frac{15 + \sqrt{57}}{3})$

Simplify each term.

Combine $- 2$ and $\frac{15 + \sqrt{57}}{3}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 - 4 \sqrt{57}}{3} \cdot (14 + \frac{- 2 (15 + \sqrt{57})}{3})$

Move the negative in front of the fraction.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 - 4 \sqrt{57}}{3} \cdot (14 - \frac{2 (15 + \sqrt{57})}{3})$

To write $14$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 - 4 \sqrt{57}}{3} \cdot (14 \cdot \frac{3}{3} - \frac{2 (15 + \sqrt{57})}{3})$

Combine fractions.

Combine $14$ and $\frac{3}{3}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 - 4 \sqrt{57}}{3} \cdot (\frac{14 \cdot 3}{3} - \frac{2 (15 + \sqrt{57})}{3})$

Combine the numerators over the common denominator.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 - 4 \sqrt{57}}{3} \cdot \frac{14 \cdot 3 - 2 (15 + \sqrt{57})}{3}$

Simplify the numerator.

Multiply $14$ by $3$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 - 4 \sqrt{57}}{3} \cdot \frac{42 - 2 (15 + \sqrt{57})}{3}$

Apply the distributive property.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 - 4 \sqrt{57}}{3} \cdot \frac{42 - 2 \cdot 15 - 2 \sqrt{57}}{3}$

Multiply $- 2$ by $15$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 - 4 \sqrt{57}}{3} \cdot \frac{42 - 30 - 2 \sqrt{57}}{3}$

Subtract $30$ from $42$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 - 4 \sqrt{57}}{3} \cdot \frac{12 - 2 \sqrt{57}}{3}$

Multiply $\frac{52 - 4 \sqrt{57}}{3} \cdot \frac{12 - 2 \sqrt{57}}{3}$ .

Multiply $\frac{52 - 4 \sqrt{57}}{3}$ by $\frac{12 - 2 \sqrt{57}}{3}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{(52 - 4 \sqrt{57}) (12 - 2 \sqrt{57})}{3 \cdot 3}$

Multiply $3$ by $3$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{(52 - 4 \sqrt{57}) (12 - 2 \sqrt{57})}{9}$

Expand $(52 - 4 \sqrt{57}) (12 - 2 \sqrt{57})$ using the FOIL Method.

Apply the distributive property.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 (12 - 2 \sqrt{57}) - 4 \sqrt{57} (12 - 2 \sqrt{57})}{9}$

Apply the distributive property.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 \cdot 12 + 52 (- 2 \sqrt{57}) - 4 \sqrt{57} (12 - 2 \sqrt{57})}{9}$

Apply the distributive property.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{52 \cdot 12 + 52 (- 2 \sqrt{57}) - 4 \sqrt{57} \cdot 12 - 4 \sqrt{57} (- 2 \sqrt{57})}{9}$

Simplify and combine like terms.

Simplify each term.

Multiply $52$ by $12$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 + 52 (- 2 \sqrt{57}) - 4 \sqrt{57} \cdot 12 - 4 \sqrt{57} (- 2 \sqrt{57})}{9}$

Multiply $- 2$ by $52$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 4 \sqrt{57} \cdot 12 - 4 \sqrt{57} (- 2 \sqrt{57})}{9}$

Multiply $12$ by $- 4$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} - 4 \sqrt{57} (- 2 \sqrt{57})}{9}$

Multiply $- 4 \sqrt{57} (- 2 \sqrt{57})$ .

Multiply $- 2$ by $- 4$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 8 \sqrt{57} \sqrt{57}}{9}$

Raise $\sqrt{57}$ to the power of $1$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 8 (\sqrt{57} \sqrt{57})}{9}$

Raise $\sqrt{57}$ to the power of $1$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 8 (\sqrt{57} \sqrt{57})}{9}$

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

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 8 {\sqrt{57}}^{1 + 1}}{9}$

Add $1$ and $1$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 8 {\sqrt{57}}^{2}}{9}$

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 8 {(57^{\frac{1}{2}})}^{2}}{9}$

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

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 8 \cdot 57^{\frac{1}{2} \cdot 2}}{9}$

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

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 8 \cdot 57^{\frac{2}{2}}}{9}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 8 \cdot 57^{\frac{2}{2}}}{9}$

Rewrite the expression.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 8 \cdot 57}{9}$

Evaluate the exponent.

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 8 \cdot 57}{9}$

Multiply $8$ by $57$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{624 - 104 \sqrt{57} - 48 \sqrt{57} + 456}{9}$

Add $624$ and $456$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{1080 - 104 \sqrt{57} - 48 \sqrt{57}}{9}$

Subtract $48 \sqrt{57}$ from $- 104 \sqrt{57}$ .

$f (\frac{15 + \sqrt{57}}{3}) = \frac{1080 - 152 \sqrt{57}}{9}$

The final answer is $\frac{1080 - 152 \sqrt{57}}{9}$ .

$y = \frac{1080 - 152 \sqrt{57}}{9}$

Step 12

Evaluate the second derivative at $x = \frac{15 - \sqrt{57}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$24 (\frac{15 - \sqrt{57}}{3}) - 120$

Step 13

Evaluate the second derivative.

Simplify each term.

Cancel the common factor of $3$ .

Factor $3$ out of $24$ .

$3 (8) \frac{15 - \sqrt{57}}{3} - 120$

Cancel the common factor.

$3 \cdot 8 \frac{15 - \sqrt{57}}{3} - 120$

Rewrite the expression.

$8 (15 - \sqrt{57}) - 120$

Apply the distributive property.

$8 \cdot 15 + 8 (- \sqrt{57}) - 120$

Multiply $8$ by $15$ .

$120 + 8 (- \sqrt{57}) - 120$

Multiply $- 1$ by $8$ .

$120 - 8 \sqrt{57} - 120$

Simplify by subtracting numbers.

Subtract $120$ from $120$ .

$0 - 8 \sqrt{57}$

Subtract $8 \sqrt{57}$ from $0$ .

$- 8 \sqrt{57}$

Step 14

$x = \frac{15 - \sqrt{57}}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{15 - \sqrt{57}}{3}$ is a local maximum

Step 15

Find the y-value when $x = \frac{15 - \sqrt{57}}{3}$ .

Replace the variable $x$ with $\frac{15 - \sqrt{57}}{3}$ in the expression.

$f (\frac{15 - \sqrt{57}}{3}) = (\frac{15 - \sqrt{57}}{3}) (16 - 2 \frac{15 - \sqrt{57}}{3}) (14 - 2 \frac{15 - \sqrt{57}}{3})$

Simplify the result.

Simplify each term.

Combine $- 2$ and $\frac{15 - \sqrt{57}}{3}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 - \sqrt{57}}{3} \cdot ((16 + \frac{- 2 (15 - \sqrt{57})}{3}) (14 - 2 \frac{15 - \sqrt{57}}{3}))$

Move the negative in front of the fraction.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 - \sqrt{57}}{3} \cdot ((16 - \frac{2 (15 - \sqrt{57})}{3}) (14 - 2 \frac{15 - \sqrt{57}}{3}))$

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 - \sqrt{57}}{3} \cdot ((16 \cdot \frac{3}{3} - \frac{2 (15 - \sqrt{57})}{3}) (14 - 2 \frac{15 - \sqrt{57}}{3}))$

Combine fractions.

Combine $16$ and $\frac{3}{3}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 - \sqrt{57}}{3} \cdot ((\frac{16 \cdot 3}{3} - \frac{2 (15 - \sqrt{57})}{3}) (14 - 2 \frac{15 - \sqrt{57}}{3}))$

Combine the numerators over the common denominator.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 - \sqrt{57}}{3} \cdot \frac{16 \cdot 3 - 2 (15 - \sqrt{57})}{3} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Simplify the numerator.

Multiply $16$ by $3$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 - \sqrt{57}}{3} \cdot \frac{48 - 2 (15 - \sqrt{57})}{3} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Apply the distributive property.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 - \sqrt{57}}{3} \cdot \frac{48 - 2 \cdot 15 - 2 (- \sqrt{57})}{3} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Multiply $- 2$ by $15$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 - \sqrt{57}}{3} \cdot \frac{48 - 30 - 2 (- \sqrt{57})}{3} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Multiply $- 1$ by $- 2$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 - \sqrt{57}}{3} \cdot \frac{48 - 30 + 2 \sqrt{57}}{3} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Subtract $30$ from $48$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 - \sqrt{57}}{3} \cdot \frac{18 + 2 \sqrt{57}}{3} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Multiply $\frac{15 - \sqrt{57}}{3} \cdot \frac{18 + 2 \sqrt{57}}{3}$ .

Multiply $\frac{15 - \sqrt{57}}{3}$ by $\frac{18 + 2 \sqrt{57}}{3}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{(15 - \sqrt{57}) (18 + 2 \sqrt{57})}{3 \cdot 3} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Multiply $3$ by $3$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{(15 - \sqrt{57}) (18 + 2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Expand $(15 - \sqrt{57}) (18 + 2 \sqrt{57})$ using the FOIL Method.

Apply the distributive property.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 (18 + 2 \sqrt{57}) - \sqrt{57} (18 + 2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Apply the distributive property.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 \cdot 18 + 15 (2 \sqrt{57}) - \sqrt{57} (18 + 2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Apply the distributive property.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{15 \cdot 18 + 15 (2 \sqrt{57}) - \sqrt{57} \cdot 18 - \sqrt{57} (2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Simplify and combine like terms.

Simplify each term.

Multiply $15$ by $18$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 15 (2 \sqrt{57}) - \sqrt{57} \cdot 18 - \sqrt{57} (2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Multiply $2$ by $15$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - \sqrt{57} \cdot 18 - \sqrt{57} (2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Multiply $18$ by $- 1$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - \sqrt{57} (2 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Multiply $- \sqrt{57} (2 \sqrt{57})$ .

Multiply $2$ by $- 1$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 2 \sqrt{57} \sqrt{57}}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Raise $\sqrt{57}$ to the power of $1$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 2 (\sqrt{57} \sqrt{57})}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Raise $\sqrt{57}$ to the power of $1$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 2 (\sqrt{57} \sqrt{57})}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

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

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 2 {\sqrt{57}}^{1 + 1}}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Add $1$ and $1$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 2 {\sqrt{57}}^{2}}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 2 {(57^{\frac{1}{2}})}^{2}}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

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

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 2 \cdot 57^{\frac{1}{2} \cdot 2}}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

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

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 2 \cdot 57^{\frac{2}{2}}}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 2 \cdot 57^{\frac{2}{2}}}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Rewrite the expression.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 2 \cdot 57}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Evaluate the exponent.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 2 \cdot 57}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Multiply $- 2$ by $57$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{270 + 30 \sqrt{57} - 18 \sqrt{57} - 114}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Subtract $114$ from $270$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{156 + 30 \sqrt{57} - 18 \sqrt{57}}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Subtract $18 \sqrt{57}$ from $30 \sqrt{57}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{156 + 12 \sqrt{57}}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Cancel the common factor of $156 + 12 \sqrt{57}$ and $9$ .

Factor $3$ out of $156$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{3 (52) + 12 \sqrt{57}}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Factor $3$ out of $12 \sqrt{57}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{3 (52) + 3 (4 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Factor $3$ out of $3 (52) + 3 (4 \sqrt{57})$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{3 (52 + 4 \sqrt{57})}{9} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Cancel the common factors.

Factor $3$ out of $9$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{3 (52 + 4 \sqrt{57})}{3 \cdot 3} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Cancel the common factor.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{3 (52 + 4 \sqrt{57})}{3 \cdot 3} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Rewrite the expression.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 + 4 \sqrt{57}}{3} \cdot (14 - 2 \frac{15 - \sqrt{57}}{3})$

Simplify each term.

Combine $- 2$ and $\frac{15 - \sqrt{57}}{3}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 + 4 \sqrt{57}}{3} \cdot (14 + \frac{- 2 (15 - \sqrt{57})}{3})$

Move the negative in front of the fraction.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 + 4 \sqrt{57}}{3} \cdot (14 - \frac{2 (15 - \sqrt{57})}{3})$

To write $14$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 + 4 \sqrt{57}}{3} \cdot (14 \cdot \frac{3}{3} - \frac{2 (15 - \sqrt{57})}{3})$

Combine fractions.

Combine $14$ and $\frac{3}{3}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 + 4 \sqrt{57}}{3} \cdot (\frac{14 \cdot 3}{3} - \frac{2 (15 - \sqrt{57})}{3})$

Combine the numerators over the common denominator.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 + 4 \sqrt{57}}{3} \cdot \frac{14 \cdot 3 - 2 (15 - \sqrt{57})}{3}$

Simplify the numerator.

Multiply $14$ by $3$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 + 4 \sqrt{57}}{3} \cdot \frac{42 - 2 (15 - \sqrt{57})}{3}$

Apply the distributive property.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 + 4 \sqrt{57}}{3} \cdot \frac{42 - 2 \cdot 15 - 2 (- \sqrt{57})}{3}$

Multiply $- 2$ by $15$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 + 4 \sqrt{57}}{3} \cdot \frac{42 - 30 - 2 (- \sqrt{57})}{3}$

Multiply $- 1$ by $- 2$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 + 4 \sqrt{57}}{3} \cdot \frac{42 - 30 + 2 \sqrt{57}}{3}$

Subtract $30$ from $42$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 + 4 \sqrt{57}}{3} \cdot \frac{12 + 2 \sqrt{57}}{3}$

Multiply $\frac{52 + 4 \sqrt{57}}{3} \cdot \frac{12 + 2 \sqrt{57}}{3}$ .

Multiply $\frac{52 + 4 \sqrt{57}}{3}$ by $\frac{12 + 2 \sqrt{57}}{3}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{(52 + 4 \sqrt{57}) (12 + 2 \sqrt{57})}{3 \cdot 3}$

Multiply $3$ by $3$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{(52 + 4 \sqrt{57}) (12 + 2 \sqrt{57})}{9}$

Expand $(52 + 4 \sqrt{57}) (12 + 2 \sqrt{57})$ using the FOIL Method.

Apply the distributive property.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 (12 + 2 \sqrt{57}) + 4 \sqrt{57} (12 + 2 \sqrt{57})}{9}$

Apply the distributive property.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 \cdot 12 + 52 (2 \sqrt{57}) + 4 \sqrt{57} (12 + 2 \sqrt{57})}{9}$

Apply the distributive property.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{52 \cdot 12 + 52 (2 \sqrt{57}) + 4 \sqrt{57} \cdot 12 + 4 \sqrt{57} (2 \sqrt{57})}{9}$

Simplify and combine like terms.

Simplify each term.

Multiply $52$ by $12$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 52 (2 \sqrt{57}) + 4 \sqrt{57} \cdot 12 + 4 \sqrt{57} (2 \sqrt{57})}{9}$

Multiply $2$ by $52$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 4 \sqrt{57} \cdot 12 + 4 \sqrt{57} (2 \sqrt{57})}{9}$

Multiply $12$ by $4$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 4 \sqrt{57} (2 \sqrt{57})}{9}$

Multiply $4 \sqrt{57} (2 \sqrt{57})$ .

Multiply $2$ by $4$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 8 \sqrt{57} \sqrt{57}}{9}$

Raise $\sqrt{57}$ to the power of $1$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 8 (\sqrt{57} \sqrt{57})}{9}$

Raise $\sqrt{57}$ to the power of $1$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 8 (\sqrt{57} \sqrt{57})}{9}$

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

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 8 {\sqrt{57}}^{1 + 1}}{9}$

Add $1$ and $1$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 8 {\sqrt{57}}^{2}}{9}$

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 8 {(57^{\frac{1}{2}})}^{2}}{9}$

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

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 8 \cdot 57^{\frac{1}{2} \cdot 2}}{9}$

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

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 8 \cdot 57^{\frac{2}{2}}}{9}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 8 \cdot 57^{\frac{2}{2}}}{9}$

Rewrite the expression.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 8 \cdot 57}{9}$

Evaluate the exponent.

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 8 \cdot 57}{9}$

Multiply $8$ by $57$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{624 + 104 \sqrt{57} + 48 \sqrt{57} + 456}{9}$

Add $624$ and $456$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{1080 + 104 \sqrt{57} + 48 \sqrt{57}}{9}$

Add $104 \sqrt{57}$ and $48 \sqrt{57}$ .

$f (\frac{15 - \sqrt{57}}{3}) = \frac{1080 + 152 \sqrt{57}}{9}$

The final answer is $\frac{1080 + 152 \sqrt{57}}{9}$ .

$y = \frac{1080 + 152 \sqrt{57}}{9}$

Step 16

These are the local extrema for $f (x) = x (16 - 2 x) (14 - 2 x)$ .

$(\frac{15 + \sqrt{57}}{3}, \frac{1080 - 152 \sqrt{57}}{9})$ is a local minima

$(\frac{15 - \sqrt{57}}{3}, \frac{1080 + 152 \sqrt{57}}{9})$ is a local maxima

Step 17

Find the Maximum/Minimum Value x(16-2x)(14-2x)

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