Find the Maximum/Minimum Value f(x)=80x-16x^2

$f (x) = 80 x - 16 x^{2}$

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 $b$ .

$x = - \frac{80}{2 (- 16)}$

Remove parentheses.

$x = - \frac{80}{2 (- 16)}$

Simplify $- \frac{80}{2 (- 16)}$ .

Cancel the common factor of $80$ and $2$ .

Factor $2$ out of $80$ .

$x = - \frac{2 \cdot 40}{2 \cdot - 16}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 40}{2 (- 16)}$

Cancel the common factor.

$x = - \frac{2 \cdot 40}{2 \cdot - 16}$

Rewrite the expression.

$x = - \frac{40}{- 16}$

Cancel the common factor of $40$ and $- 16$ .

Factor $8$ out of $40$ .

$x = - \frac{8 (5)}{- 16}$

Cancel the common factors.

Factor $8$ out of $- 16$ .

$x = - \frac{8 \cdot 5}{8 \cdot - 2}$

Cancel the common factor.

$x = - \frac{8 \cdot 5}{8 \cdot - 2}$

Rewrite the expression.

$x = - \frac{5}{- 2}$

Move the negative in front of the fraction.

$x = - - \frac{5}{2}$

Multiply $- - \frac{5}{2}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{5}{2})$

Multiply $\frac{5}{2}$ by $1$ .

$x = \frac{5}{2}$

Step 3

Evaluate $f (\frac{5}{2})$ .

Replace the variable $x$ with $\frac{5}{2}$ in the expression.

$f (\frac{5}{2}) = 80 (\frac{5}{2}) - 16 {(\frac{5}{2})}^{2}$

Simplify the result.

Simplify each term.

Cancel the common factor of $2$ .

Factor $2$ out of $80$ .

$f (\frac{5}{2}) = 2 (40) (\frac{5}{2}) - 16 {(\frac{5}{2})}^{2}$

Cancel the common factor.

$f (\frac{5}{2}) = 2 \cdot (40 (\frac{5}{2})) - 16 {(\frac{5}{2})}^{2}$

Rewrite the expression.

$f (\frac{5}{2}) = 40 \cdot 5 - 16 {(\frac{5}{2})}^{2}$

Multiply $40$ by $5$ .

$f (\frac{5}{2}) = 200 - 16 {(\frac{5}{2})}^{2}$

Apply the product rule to $\frac{5}{2}$ .

$f (\frac{5}{2}) = 200 - 16 \frac{5^{2}}{2^{2}}$

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

$f (\frac{5}{2}) = 200 - 16 \frac{25}{2^{2}}$

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

$f (\frac{5}{2}) = 200 - 16 (\frac{25}{4})$

Cancel the common factor of $4$ .

Factor $4$ out of $- 16$ .

$f (\frac{5}{2}) = 200 + 4 (- 4) (\frac{25}{4})$

Cancel the common factor.

$f (\frac{5}{2}) = 200 + 4 \cdot (- 4 (\frac{25}{4}))$

Rewrite the expression.

$f (\frac{5}{2}) = 200 - 4 \cdot 25$

Multiply $- 4$ by $25$ .

$f (\frac{5}{2}) = 200 - 100$

Subtract $100$ from $200$ .

$f (\frac{5}{2}) = 100$

The final answer is $100$ .

$100$

Step 4

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

$(\frac{5}{2}, 100)$

Step 5

Find the Maximum/Minimum Value f(x)=80x-16x^2

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