Find the Maximum/Minimum Value f(x)=x^2+11x

$f (x) = x^{2} + 11 x$

Step 1

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

$f_{min}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{11}{2 (1)}$

Remove parentheses.

$x = - \frac{11}{2 (1)}$

Multiply $2$ by $1$ .

$x = - \frac{11}{2}$

Step 3

Evaluate $f (- \frac{11}{2})$ .

Replace the variable $x$ with $- \frac{11}{2}$ in the expression.

$f (- \frac{11}{2}) = {(- \frac{11}{2})}^{2} + 11 (- \frac{11}{2})$

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 $- \frac{11}{2}$ .

$f (- \frac{11}{2}) = {(- 1)}^{2} {(\frac{11}{2})}^{2} + 11 (- \frac{11}{2})$

Apply the product rule to $\frac{11}{2}$ .

$f (- \frac{11}{2}) = {(- 1)}^{2} (\frac{11^{2}}{2^{2}}) + 11 (- \frac{11}{2})$

Raise $- 1$ to the power of $2$ .

$f (- \frac{11}{2}) = 1 (\frac{11^{2}}{2^{2}}) + 11 (- \frac{11}{2})$

Multiply $\frac{11^{2}}{2^{2}}$ by $1$ .

$f (- \frac{11}{2}) = \frac{11^{2}}{2^{2}} + 11 (- \frac{11}{2})$

Raise $11$ to the power of $2$ .

$f (- \frac{11}{2}) = \frac{121}{2^{2}} + 11 (- \frac{11}{2})$

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

$f (- \frac{11}{2}) = \frac{121}{4} + 11 (- \frac{11}{2})$

Multiply $11 (- \frac{11}{2})$ .

Multiply $- 1$ by $11$ .

$f (- \frac{11}{2}) = \frac{121}{4} - 11 (\frac{11}{2})$

Combine $- 11$ and $\frac{11}{2}$ .

$f (- \frac{11}{2}) = \frac{121}{4} + \frac{- 11 \cdot 11}{2}$

Multiply $- 11$ by $11$ .

$f (- \frac{11}{2}) = \frac{121}{4} + \frac{- 121}{2}$

Move the negative in front of the fraction.

$f (- \frac{11}{2}) = \frac{121}{4} - \frac{121}{2}$

To write $- \frac{121}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{11}{2}) = \frac{121}{4} - \frac{121}{2} \cdot \frac{2}{2}$

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

Multiply $\frac{121}{2}$ by $\frac{2}{2}$ .

$f (- \frac{11}{2}) = \frac{121}{4} - \frac{121 \cdot 2}{2 \cdot 2}$

Multiply $2$ by $2$ .

$f (- \frac{11}{2}) = \frac{121}{4} - \frac{121 \cdot 2}{4}$

Combine the numerators over the common denominator.

$f (- \frac{11}{2}) = \frac{121 - 121 \cdot 2}{4}$

Simplify the numerator.

Multiply $- 121$ by $2$ .

$f (- \frac{11}{2}) = \frac{121 - 242}{4}$

Subtract $242$ from $121$ .

$f (- \frac{11}{2}) = \frac{- 121}{4}$

Move the negative in front of the fraction.

$f (- \frac{11}{2}) = - \frac{121}{4}$

The final answer is $- \frac{121}{4}$ .

$- \frac{121}{4}$

Step 4

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

$(- \frac{11}{2}, - \frac{121}{4})$

Step 5

Find the Maximum/Minimum Value f(x)=x^2+11x

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