Find the Maximum/Minimum Value f(x)=68x-16x^2

$f (x) = 68 x - 16 x^{2}$

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 $b$ .

$x = - \frac{68}{2 (- 16)}$

Remove parentheses.

$x = - \frac{68}{2 (- 16)}$

Simplify $- \frac{68}{2 (- 16)}$ .

Cancel the common factor of $68$ and $2$ .

Factor $2$ out of $68$ .

$x = - \frac{2 \cdot 34}{2 \cdot - 16}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 34}{2 (- 16)}$

Cancel the common factor.

$x = - \frac{2 \cdot 34}{2 \cdot - 16}$

Rewrite the expression.

$x = - \frac{34}{- 16}$

Cancel the common factor of $34$ and $- 16$ .

Factor $2$ out of $34$ .

$x = - \frac{2 (17)}{- 16}$

Cancel the common factors.

Factor $2$ out of $- 16$ .

$x = - \frac{2 \cdot 17}{2 \cdot - 8}$

Cancel the common factor.

$x = - \frac{2 \cdot 17}{2 \cdot - 8}$

Rewrite the expression.

$x = - \frac{17}{- 8}$

Move the negative in front of the fraction.

$x = - - \frac{17}{8}$

Multiply $- - \frac{17}{8}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{17}{8})$

Multiply $\frac{17}{8}$ by $1$ .

$x = \frac{17}{8}$

Step 3

Evaluate $f (\frac{17}{8})$ .

Replace the variable $x$ with $\frac{17}{8}$ in the expression.

$f (\frac{17}{8}) = 68 (\frac{17}{8}) - 16 {(\frac{17}{8})}^{2}$

Simplify the result.

Simplify each term.

Cancel the common factor of $4$ .

Factor $4$ out of $68$ .

$f (\frac{17}{8}) = 4 (17) (\frac{17}{8}) - 16 {(\frac{17}{8})}^{2}$

Factor $4$ out of $8$ .

$f (\frac{17}{8}) = 4 \cdot (17 (\frac{17}{4 \cdot 2})) - 16 {(\frac{17}{8})}^{2}$

Cancel the common factor.

$f (\frac{17}{8}) = 4 \cdot (17 (\frac{17}{4 \cdot 2})) - 16 {(\frac{17}{8})}^{2}$

Rewrite the expression.

$f (\frac{17}{8}) = 17 (\frac{17}{2}) - 16 {(\frac{17}{8})}^{2}$

Combine $17$ and $\frac{17}{2}$ .

$f (\frac{17}{8}) = \frac{17 \cdot 17}{2} - 16 {(\frac{17}{8})}^{2}$

Multiply $17$ by $17$ .

$f (\frac{17}{8}) = \frac{289}{2} - 16 {(\frac{17}{8})}^{2}$

Apply the product rule to $\frac{17}{8}$ .

$f (\frac{17}{8}) = \frac{289}{2} - 16 \frac{17^{2}}{8^{2}}$

Raise $17$ to the power of $2$ .

$f (\frac{17}{8}) = \frac{289}{2} - 16 \frac{289}{8^{2}}$

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

$f (\frac{17}{8}) = \frac{289}{2} - 16 (\frac{289}{64})$

Cancel the common factor of $16$ .

Factor $16$ out of $- 16$ .

$f (\frac{17}{8}) = \frac{289}{2} + 16 (- 1) (\frac{289}{64})$

Factor $16$ out of $64$ .

$f (\frac{17}{8}) = \frac{289}{2} + 16 \cdot (- 1 \frac{289}{16 \cdot 4})$

Cancel the common factor.

$f (\frac{17}{8}) = \frac{289}{2} + 16 \cdot (- 1 \frac{289}{16 \cdot 4})$

Rewrite the expression.

$f (\frac{17}{8}) = \frac{289}{2} - 1 (\frac{289}{4})$

Rewrite $- 1 (\frac{289}{4})$ as $- (\frac{289}{4})$ .

$f (\frac{17}{8}) = \frac{289}{2} - \frac{289}{4}$

To write $\frac{289}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{17}{8}) = \frac{289}{2} \cdot \frac{2}{2} - \frac{289}{4}$

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

Multiply $\frac{289}{2}$ by $\frac{2}{2}$ .

$f (\frac{17}{8}) = \frac{289 \cdot 2}{2 \cdot 2} - \frac{289}{4}$

Multiply $2$ by $2$ .

$f (\frac{17}{8}) = \frac{289 \cdot 2}{4} - \frac{289}{4}$

Combine the numerators over the common denominator.

$f (\frac{17}{8}) = \frac{289 \cdot 2 - 289}{4}$

Simplify the numerator.

Multiply $289$ by $2$ .

$f (\frac{17}{8}) = \frac{578 - 289}{4}$

Subtract $289$ from $578$ .

$f (\frac{17}{8}) = \frac{289}{4}$

The final answer is $\frac{289}{4}$ .

$\frac{289}{4}$

Step 4

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

$(\frac{17}{8}, \frac{289}{4})$

Step 5

Find the Maximum/Minimum Value f(x)=68x-16x^2

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