Find the Maximum/Minimum Value f(x)=-x^2-7x-7

$f (x) = - x^{2} - 7 x - 7$

The maximum or minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is negative, the maximum value of the function is $f (- \frac{b}{2 a})$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{max}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Find the value of $x$ equal to $- \frac{b}{2 a}$ .

$x = - \frac{b}{2 a}$

Substitute in the values of $a$ and $b$ .

$x = - \frac{- 7}{2 (- 1)}$

Remove parentheses.

$x = - \frac{- 7}{2 (- 1)}$

Simplify $- \frac{- 7}{2 (- 1)}$ .

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Multiply $2$ by $- 1$ .

$x = - \frac{- 7}{- 2}$

Dividing two negative values results in a positive value.

$x = - \frac{7}{2}$

Replace the variable $x$ with $- \frac{7}{2}$ in the expression.

$f (- \frac{7}{2}) = - {(- \frac{7}{2})}^{2} - 7 (- \frac{7}{2}) - 7$

Simplify the result.

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Simplify each term.

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

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Apply the product rule to $- \frac{7}{2}$ .

$f (- \frac{7}{2}) = - ({(- 1)}^{2} {(\frac{7}{2})}^{2}) - 7 (- \frac{7}{2}) - 7$

Apply the product rule to $\frac{7}{2}$ .

$f (- \frac{7}{2}) = - ({(- 1)}^{2} (\frac{7^{2}}{2^{2}})) - 7 (- \frac{7}{2}) - 7$

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

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

$f (- \frac{7}{2}) = {(- 1)}^{2} \cdot (- 1 \frac{7^{2}}{2^{2}}) - 7 (- \frac{7}{2}) - 7$

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

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

$f (- \frac{7}{2}) = {(- 1)}^{2} \cdot ((- 1) (\frac{7^{2}}{2^{2}})) - 7 (- \frac{7}{2}) - 7$

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

$f (- \frac{7}{2}) = {(- 1)}^{2 + 1} (\frac{7^{2}}{2^{2}}) - 7 (- \frac{7}{2}) - 7$

Add $2$ and $1$ .

$f (- \frac{7}{2}) = {(- 1)}^{3} (\frac{7^{2}}{2^{2}}) - 7 (- \frac{7}{2}) - 7$

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

$f (- \frac{7}{2}) = - \frac{7^{2}}{2^{2}} - 7 (- \frac{7}{2}) - 7$

Raise $7$ to the power of $2$ .

$f (- \frac{7}{2}) = - \frac{49}{2^{2}} - 7 (- \frac{7}{2}) - 7$

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

$f (- \frac{7}{2}) = - \frac{49}{4} - 7 (- \frac{7}{2}) - 7$

Multiply $- 7 (- \frac{7}{2})$ .

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Multiply $- 1$ by $- 7$ .

$f (- \frac{7}{2}) = - \frac{49}{4} + 7 (\frac{7}{2}) - 7$

Combine $7$ and $\frac{7}{2}$ .

$f (- \frac{7}{2}) = - \frac{49}{4} + \frac{7 \cdot 7}{2} - 7$

Multiply $7$ by $7$ .

$f (- \frac{7}{2}) = - \frac{49}{4} + \frac{49}{2} - 7$

Find the common denominator.

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Multiply $\frac{49}{2}$ by $\frac{2}{2}$ .

$f (- \frac{7}{2}) = - \frac{49}{4} + \frac{49}{2} \cdot \frac{2}{2} - 7$

Combine.

$f (- \frac{7}{2}) = - \frac{49}{4} + \frac{49 \cdot 2}{2 \cdot 2} - 7$

Write $- 7$ as a fraction with denominator $1$ .

$f (- \frac{7}{2}) = - \frac{49}{4} + \frac{49 \cdot 2}{2 \cdot 2} + \frac{- 7}{1}$

Multiply $\frac{- 7}{1}$ by $\frac{4}{4}$ .

$f (- \frac{7}{2}) = - \frac{49}{4} + \frac{49 \cdot 2}{2 \cdot 2} + \frac{- 7}{1} \cdot \frac{4}{4}$

Multiply $\frac{- 7}{1}$ and $\frac{4}{4}$ .

$f (- \frac{7}{2}) = - \frac{49}{4} + \frac{49 \cdot 2}{2 \cdot 2} + \frac{- 7 \cdot 4}{4}$

Multiply $2$ by $2$ .

$f (- \frac{7}{2}) = - \frac{49}{4} + \frac{49 \cdot 2}{4} + \frac{- 7 \cdot 4}{4}$

Combine fractions.

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Combine fractions with similar denominators.

$f (- \frac{7}{2}) = \frac{- 49 + 49 \cdot 2 - 7 \cdot 4}{4}$

Multiply.

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Multiply $49$ by $2$ .

$f (- \frac{7}{2}) = \frac{- 49 + 98 - 7 \cdot 4}{4}$

Multiply $- 7$ by $4$ .

$f (- \frac{7}{2}) = \frac{- 49 + 98 - 28}{4}$

Simplify the numerator.

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Add $- 49$ and $98$ .

$f (- \frac{7}{2}) = \frac{49 - 28}{4}$

Subtract $28$ from $49$ .

$f (- \frac{7}{2}) = \frac{21}{4}$

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

$\frac{21}{4}$

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

$(- \frac{7}{2}, \frac{21}{4})$

Find the Maximum/Minimum Value f(x)=-x^2-7x-7

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