Find the Maximum/Minimum Value f(x)=-4x^2+16x

$f (x) = - 4 x^{2} + 16 x$

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 $b$ .

$x = - \frac{16}{2 (- 4)}$

Remove parentheses.

$x = - \frac{16}{2 (- 4)}$

Simplify $- \frac{16}{2 (- 4)}$ .

Cancel the common factor of $16$ and $2$ .

Factor $2$ out of $16$ .

$x = - \frac{2 \cdot 8}{2 \cdot - 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 4$ .

$x = - \frac{2 \cdot 8}{2 (- 4)}$

Cancel the common factor.

$x = - \frac{2 \cdot 8}{2 \cdot - 4}$

Rewrite the expression.

$x = - \frac{8}{- 4}$

Cancel the common factor of $8$ and $- 4$ .

Factor $4$ out of $8$ .

$x = - \frac{4 (2)}{- 4}$

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$x = - (- 1 \cdot 2)$

Multiply $- (- 1 \cdot 2)$ .

Multiply $- 1$ by $2$ .

$x = - - 2$

Multiply $- 1$ by $- 2$ .

$x = 2$

Step 3

Evaluate $f (2)$ .

Replace the variable $x$ with $2$ in the expression.

$f (2) = - 4 {(2)}^{2} + 16 (2)$

Simplify the result.

Simplify each term.

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

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

Multiply $- 4$ by $4$ .

$f (2) = - 16 + 16 (2)$

Multiply $16$ by $2$ .

$f (2) = - 16 + 32$

Add $- 16$ and $32$ .

$f (2) = 16$

The final answer is $16$ .

$16$

Step 4

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

$(2, 16)$

Step 5

Find the Maximum/Minimum Value f(x)=-4x^2+16x

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