Find the Maximum/Minimum Value f(x)=-x^2-16x-64

$f (x) = - x^{2} - 16 x - 64$

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 $b$ .

$x = - \frac{- 16}{2 (- 1)}$

Remove parentheses.

$x = - \frac{- 16}{2 (- 1)}$

Simplify $- \frac{- 16}{2 (- 1)}$ .

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

Factor $2$ out of $- 16$ .

$x = - \frac{2 \cdot - 8}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{- 8}{- 1}$ .

$x = - (- 1 \cdot - 8)$

Rewrite $- 1 \cdot - 8$ as $- - 8$ .

$x = - - - 8$

Multiply $- - - 8$ .

Multiply $- 1$ by $- 8$ .

$x = - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$x = - 8$

Step 3

Evaluate $f (- 8)$ .

Replace the variable $x$ with $- 8$ in the expression.

$f (- 8) = - {(- 8)}^{2} - 16 \cdot - 8 - 64$

Simplify the result.

Simplify each term.

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

$f (- 8) = - 1 \cdot 64 - 16 \cdot - 8 - 64$

Multiply $- 1$ by $64$ .

$f (- 8) = - 64 - 16 \cdot - 8 - 64$

Multiply $- 16$ by $- 8$ .

$f (- 8) = - 64 + 128 - 64$

Simplify by adding and subtracting.

Add $- 64$ and $128$ .

$f (- 8) = 64 - 64$

Subtract $64$ from $64$ .

$f (- 8) = 0$

The final answer is $0$ .

$0$

Step 4

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

$(- 8, 0)$

Step 5

Find the Maximum/Minimum Value f(x)=-x^2-16x-64

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