Find the Maximum/Minimum Value 116x-16x^2

$116 x - 16 x^{2}$

Step 1

The maximum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is negative, the maximum value of the function is $f (- \frac{b}{2 a})$ .

$f_{max}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 2

Find the value of $x = - \frac{b}{2 a}$ .

Substitute in the values of $a$ and $b$ .

$x = - \frac{116}{2 (- 16)}$

Remove parentheses.

$x = - \frac{116}{2 (- 16)}$

Simplify $- \frac{116}{2 (- 16)}$ .

Cancel the common factor of $116$ and $2$ .

Factor $2$ out of $116$ .

$x = - \frac{2 \cdot 58}{2 \cdot - 16}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 16$ .

$x = - \frac{2 \cdot 58}{2 (- 16)}$

Cancel the common factor.

$x = - \frac{2 \cdot 58}{2 \cdot - 16}$

Rewrite the expression.

$x = - \frac{58}{- 16}$

Cancel the common factor of $58$ and $- 16$ .

Factor $2$ out of $58$ .

$x = - \frac{2 (29)}{- 16}$

Cancel the common factors.

Factor $2$ out of $- 16$ .

$x = - \frac{2 \cdot 29}{2 \cdot - 8}$

Cancel the common factor.

$x = - \frac{2 \cdot 29}{2 \cdot - 8}$

Rewrite the expression.

$x = - \frac{29}{- 8}$

Move the negative in front of the fraction.

$x = - - \frac{29}{8}$

Multiply $- - \frac{29}{8}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{29}{8})$

Multiply $\frac{29}{8}$ by $1$ .

$x = \frac{29}{8}$

Step 3

Evaluate $f (\frac{29}{8})$ .

Replace the variable $x$ with $\frac{29}{8}$ in the expression.

$f (\frac{29}{8}) = 116 (\frac{29}{8}) - 16 {(\frac{29}{8})}^{2}$

Simplify the result.

Simplify each term.

Cancel the common factor of $4$ .

Factor $4$ out of $116$ .

$f (\frac{29}{8}) = 4 (29) (\frac{29}{8}) - 16 {(\frac{29}{8})}^{2}$

Factor $4$ out of $8$ .

$f (\frac{29}{8}) = 4 \cdot (29 (\frac{29}{4 \cdot 2})) - 16 {(\frac{29}{8})}^{2}$

Cancel the common factor.

$f (\frac{29}{8}) = 4 \cdot (29 (\frac{29}{4 \cdot 2})) - 16 {(\frac{29}{8})}^{2}$

Rewrite the expression.

$f (\frac{29}{8}) = 29 (\frac{29}{2}) - 16 {(\frac{29}{8})}^{2}$

Combine $29$ and $\frac{29}{2}$ .

$f (\frac{29}{8}) = \frac{29 \cdot 29}{2} - 16 {(\frac{29}{8})}^{2}$

Multiply $29$ by $29$ .

$f (\frac{29}{8}) = \frac{841}{2} - 16 {(\frac{29}{8})}^{2}$

Apply the product rule to $\frac{29}{8}$ .

$f (\frac{29}{8}) = \frac{841}{2} - 16 \frac{29^{2}}{8^{2}}$

Raise $29$ to the power of $2$ .

$f (\frac{29}{8}) = \frac{841}{2} - 16 \frac{841}{8^{2}}$

Raise $8$ to the power of $2$ .

$f (\frac{29}{8}) = \frac{841}{2} - 16 (\frac{841}{64})$

Cancel the common factor of $16$ .

Factor $16$ out of $- 16$ .

$f (\frac{29}{8}) = \frac{841}{2} + 16 (- 1) (\frac{841}{64})$

Factor $16$ out of $64$ .

$f (\frac{29}{8}) = \frac{841}{2} + 16 \cdot (- 1 \frac{841}{16 \cdot 4})$

Cancel the common factor.

$f (\frac{29}{8}) = \frac{841}{2} + 16 \cdot (- 1 \frac{841}{16 \cdot 4})$

Rewrite the expression.

$f (\frac{29}{8}) = \frac{841}{2} - 1 (\frac{841}{4})$

Rewrite $- 1 (\frac{841}{4})$ as $- (\frac{841}{4})$ .

$f (\frac{29}{8}) = \frac{841}{2} - \frac{841}{4}$

To write $\frac{841}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{29}{8}) = \frac{841}{2} \cdot \frac{2}{2} - \frac{841}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{841}{2}$ by $\frac{2}{2}$ .

$f (\frac{29}{8}) = \frac{841 \cdot 2}{2 \cdot 2} - \frac{841}{4}$

Multiply $2$ by $2$ .

$f (\frac{29}{8}) = \frac{841 \cdot 2}{4} - \frac{841}{4}$

Combine the numerators over the common denominator.

$f (\frac{29}{8}) = \frac{841 \cdot 2 - 841}{4}$

Simplify the numerator.

Multiply $841$ by $2$ .

$f (\frac{29}{8}) = \frac{1682 - 841}{4}$

Subtract $841$ from $1682$ .

$f (\frac{29}{8}) = \frac{841}{4}$

The final answer is $\frac{841}{4}$ .

$\frac{841}{4}$

Step 4

Use the $x$ and $y$ values to find where the maximum occurs.

$(\frac{29}{8}, \frac{841}{4})$

Step 5

Find the Maximum/Minimum Value 116x-16x^2

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