Find the Maximum/Minimum Value f(x)=-2x^2+16x-9

$f (x) = - 2 x^{2} + 16 x - 9$

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 $b$ .

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

Remove parentheses.

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

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

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

Factor $2$ out of $16$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $8$ and $- 2$ .

Factor $2$ out of $8$ .

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

Move the negative one from the denominator of $\frac{4}{- 1}$ .

$x = - (- 1 \cdot 4)$

Multiply $- (- 1 \cdot 4)$ .

Multiply $- 1$ by $4$ .

$x = - - 4$

Multiply $- 1$ by $- 4$ .

$x = 4$

Step 3

Evaluate $f (4)$ .

Replace the variable $x$ with $4$ in the expression.

$f (4) = - 2 {(4)}^{2} + 16 (4) - 9$

Simplify the result.

Simplify each term.

Raise $4$ to the power of $2$ .

$f (4) = - 2 \cdot 16 + 16 (4) - 9$

Multiply $- 2$ by $16$ .

$f (4) = - 32 + 16 (4) - 9$

Multiply $16$ by $4$ .

$f (4) = - 32 + 64 - 9$

Simplify by adding and subtracting.

Add $- 32$ and $64$ .

$f (4) = 32 - 9$

Subtract $9$ from $32$ .

$f (4) = 23$

The final answer is $23$ .

$23$

Step 4

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

$(4, 23)$

Step 5

Find the Maximum/Minimum Value f(x)=-2x^2+16x-9

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