Find the Maximum/Minimum Value f(x)=-5x-x^2

$f (x) = - 5 x - x^{2}$

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 $b$ .

$x = - \frac{- 5}{2 (- 1)}$

Remove parentheses.

$x = - \frac{- 5}{2 (- 1)}$

Simplify $- \frac{- 5}{2 (- 1)}$ .

Multiply $2$ by $- 1$ .

$x = - \frac{- 5}{- 2}$

Dividing two negative values results in a positive value.

$x = - \frac{5}{2}$

Step 3

Evaluate $f (- \frac{5}{2})$ .

Replace the variable $x$ with $- \frac{5}{2}$ in the expression.

$f (- \frac{5}{2}) = - 5 (- \frac{5}{2}) - {(- \frac{5}{2})}^{2}$

Simplify the result.

Simplify each term.

Multiply $- 5 (- \frac{5}{2})$ .

Multiply $- 1$ by $- 5$ .

$f (- \frac{5}{2}) = 5 (\frac{5}{2}) - {(- \frac{5}{2})}^{2}$

Combine $5$ and $\frac{5}{2}$ .

$f (- \frac{5}{2}) = \frac{5 \cdot 5}{2} - {(- \frac{5}{2})}^{2}$

Multiply $5$ by $5$ .

$f (- \frac{5}{2}) = \frac{25}{2} - {(- \frac{5}{2})}^{2}$

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

Apply the product rule to $- \frac{5}{2}$ .

$f (- \frac{5}{2}) = \frac{25}{2} - ({(- 1)}^{2} {(\frac{5}{2})}^{2})$

Apply the product rule to $\frac{5}{2}$ .

$f (- \frac{5}{2}) = \frac{25}{2} - ({(- 1)}^{2} (\frac{5^{2}}{2^{2}}))$

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

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

$f (- \frac{5}{2}) = \frac{25}{2} + {(- 1)}^{2} \cdot (- 1 \frac{5^{2}}{2^{2}})$

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

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

$f (- \frac{5}{2}) = \frac{25}{2} + {(- 1)}^{2} \cdot ((- 1) (\frac{5^{2}}{2^{2}}))$

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

$f (- \frac{5}{2}) = \frac{25}{2} + {(- 1)}^{2 + 1} (\frac{5^{2}}{2^{2}})$

Add $2$ and $1$ .

$f (- \frac{5}{2}) = \frac{25}{2} + {(- 1)}^{3} (\frac{5^{2}}{2^{2}})$

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

$f (- \frac{5}{2}) = \frac{25}{2} - \frac{5^{2}}{2^{2}}$

Raise $5$ to the power of $2$ .

$f (- \frac{5}{2}) = \frac{25}{2} - \frac{25}{2^{2}}$

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

$f (- \frac{5}{2}) = \frac{25}{2} - \frac{25}{4}$

To write $\frac{25}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = \frac{25}{2} \cdot \frac{2}{2} - \frac{25}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{25}{2}$ by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = \frac{25 \cdot 2}{2 \cdot 2} - \frac{25}{4}$

Multiply $2$ by $2$ .

$f (- \frac{5}{2}) = \frac{25 \cdot 2}{4} - \frac{25}{4}$

Combine the numerators over the common denominator.

$f (- \frac{5}{2}) = \frac{25 \cdot 2 - 25}{4}$

Simplify the numerator.

Multiply $25$ by $2$ .

$f (- \frac{5}{2}) = \frac{50 - 25}{4}$

Subtract $25$ from $50$ .

$f (- \frac{5}{2}) = \frac{25}{4}$

The final answer is $\frac{25}{4}$ .

$\frac{25}{4}$

Step 4

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

$(- \frac{5}{2}, \frac{25}{4})$

Step 5

Find the Maximum/Minimum Value f(x)=-5x-x^2

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