Find the Maximum/Minimum Value m(x)=22x-x^2

$m (x) = 22 x - x^{2}$

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 $b$ .

$x = - \frac{22}{2 (- 1)}$

Remove parentheses.

$x = - \frac{22}{2 (- 1)}$

Simplify $- \frac{22}{2 (- 1)}$ .

Cancel the common factor of $22$ and $2$ .

Factor $2$ out of $22$ .

$x = - \frac{2 \cdot 11}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{11}{- 1}$ .

$x = - (- 1 \cdot 11)$

Multiply $- (- 1 \cdot 11)$ .

Multiply $- 1$ by $11$ .

$x = - - 11$

Multiply $- 1$ by $- 11$ .

$x = 11$

Step 3

Evaluate $f (11)$ .

Replace the variable $x$ with $11$ in the expression.

$m (11) = 22 (11) - {(11)}^{2}$

Simplify the result.

Simplify each term.

Multiply $22$ by $11$ .

$m (11) = 242 - {(11)}^{2}$

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

$m (11) = 242 - 1 \cdot 121$

Multiply $- 1$ by $121$ .

$m (11) = 242 - 121$

Subtract $121$ from $242$ .

$m (11) = 121$

The final answer is $121$ .

$121$

Step 4

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

$(11, 121)$

Step 5

Find the Maximum/Minimum Value m(x)=22x-x^2

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