Find the Maximum/Minimum Value f(x)=-x^2-9x-15

$f (x) = - x^{2} - 9 x - 15$

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 $b$ .

$x = - \frac{- 9}{2 (- 1)}$

Remove parentheses.

$x = - \frac{- 9}{2 (- 1)}$

Simplify $- \frac{- 9}{2 (- 1)}$ .

Multiply $2$ by $- 1$ .

$x = - \frac{- 9}{- 2}$

Dividing two negative values results in a positive value.

$x = - \frac{9}{2}$

Step 3

Evaluate $f (- \frac{9}{2})$ .

Replace the variable $x$ with $- \frac{9}{2}$ in the expression.

$f (- \frac{9}{2}) = - {(- \frac{9}{2})}^{2} - 9 (- \frac{9}{2}) - 15$

Simplify the result.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{9}{2}$ .

$f (- \frac{9}{2}) = - ({(- 1)}^{2} {(\frac{9}{2})}^{2}) - 9 (- \frac{9}{2}) - 15$

Apply the product rule to $\frac{9}{2}$ .

$f (- \frac{9}{2}) = - ({(- 1)}^{2} (\frac{9^{2}}{2^{2}})) - 9 (- \frac{9}{2}) - 15$

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Move ${(- 1)}^{2}$ .

$f (- \frac{9}{2}) = {(- 1)}^{2} \cdot (- 1 \frac{9^{2}}{2^{2}}) - 9 (- \frac{9}{2}) - 15$

Multiply ${(- 1)}^{2}$ by $- 1$ .

Raise $- 1$ to the power of $1$ .

$f (- \frac{9}{2}) = {(- 1)}^{2} \cdot ((- 1) (\frac{9^{2}}{2^{2}})) - 9 (- \frac{9}{2}) - 15$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{9}{2}) = {(- 1)}^{2 + 1} (\frac{9^{2}}{2^{2}}) - 9 (- \frac{9}{2}) - 15$

Add $2$ and $1$ .

$f (- \frac{9}{2}) = {(- 1)}^{3} (\frac{9^{2}}{2^{2}}) - 9 (- \frac{9}{2}) - 15$

Raise $- 1$ to the power of $3$ .

$f (- \frac{9}{2}) = - \frac{9^{2}}{2^{2}} - 9 (- \frac{9}{2}) - 15$

Raise $9$ to the power of $2$ .

$f (- \frac{9}{2}) = - \frac{81}{2^{2}} - 9 (- \frac{9}{2}) - 15$

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

$f (- \frac{9}{2}) = - \frac{81}{4} - 9 (- \frac{9}{2}) - 15$

Multiply $- 9 (- \frac{9}{2})$ .

Multiply $- 1$ by $- 9$ .

$f (- \frac{9}{2}) = - \frac{81}{4} + 9 (\frac{9}{2}) - 15$

Combine $9$ and $\frac{9}{2}$ .

$f (- \frac{9}{2}) = - \frac{81}{4} + \frac{9 \cdot 9}{2} - 15$

Multiply $9$ by $9$ .

$f (- \frac{9}{2}) = - \frac{81}{4} + \frac{81}{2} - 15$

Find the common denominator.

Multiply $\frac{81}{2}$ by $\frac{2}{2}$ .

$f (- \frac{9}{2}) = - \frac{81}{4} + \frac{81}{2} \cdot \frac{2}{2} - 15$

Multiply $\frac{81}{2}$ by $\frac{2}{2}$ .

$f (- \frac{9}{2}) = - \frac{81}{4} + \frac{81 \cdot 2}{2 \cdot 2} - 15$

Write $- 15$ as a fraction with denominator $1$ .

$f (- \frac{9}{2}) = - \frac{81}{4} + \frac{81 \cdot 2}{2 \cdot 2} + \frac{- 15}{1}$

Multiply $\frac{- 15}{1}$ by $\frac{4}{4}$ .

$f (- \frac{9}{2}) = - \frac{81}{4} + \frac{81 \cdot 2}{2 \cdot 2} + \frac{- 15}{1} \cdot \frac{4}{4}$

Multiply $\frac{- 15}{1}$ by $\frac{4}{4}$ .

$f (- \frac{9}{2}) = - \frac{81}{4} + \frac{81 \cdot 2}{2 \cdot 2} + \frac{- 15 \cdot 4}{4}$

Multiply $2$ by $2$ .

$f (- \frac{9}{2}) = - \frac{81}{4} + \frac{81 \cdot 2}{4} + \frac{- 15 \cdot 4}{4}$

Combine the numerators over the common denominator.

$f (- \frac{9}{2}) = \frac{- 81 + 81 \cdot 2 - 15 \cdot 4}{4}$

Simplify each term.

Multiply $81$ by $2$ .

$f (- \frac{9}{2}) = \frac{- 81 + 162 - 15 \cdot 4}{4}$

Multiply $- 15$ by $4$ .

$f (- \frac{9}{2}) = \frac{- 81 + 162 - 60}{4}$

Simplify by adding and subtracting.

Add $- 81$ and $162$ .

$f (- \frac{9}{2}) = \frac{81 - 60}{4}$

Subtract $60$ from $81$ .

$f (- \frac{9}{2}) = \frac{21}{4}$

The final answer is $\frac{21}{4}$ .

$\frac{21}{4}$

Step 4

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

$(- \frac{9}{2}, \frac{21}{4})$

Step 5

Find the Maximum/Minimum Value f(x)=-x^2-9x-15

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