Find the Maximum/Minimum Value m(x)=10x-x^2

$m (x) = 10 x - x^{2}$

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 $b$ .

$x = - \frac{10}{2 (- 1)}$

Remove parentheses.

$x = - \frac{10}{2 (- 1)}$

Simplify $- \frac{10}{2 (- 1)}$ .

Cancel the common factor of $10$ and $2$ .

Factor $2$ out of $10$ .

$x = - \frac{2 \cdot 5}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$x = - (- 1 \cdot 5)$

Multiply $- (- 1 \cdot 5)$ .

Multiply $- 1$ by $5$ .

$x = - - 5$

Multiply $- 1$ by $- 5$ .

$x = 5$

Step 3

Evaluate $f (5)$ .

Replace the variable $x$ with $5$ in the expression.

$m (5) = 10 (5) - {(5)}^{2}$

Simplify the result.

Simplify each term.

Multiply $10$ by $5$ .

$m (5) = 50 - {(5)}^{2}$

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

$m (5) = 50 - 1 \cdot 25$

Multiply $- 1$ by $25$ .

$m (5) = 50 - 25$

Subtract $25$ from $50$ .

$m (5) = 25$

The final answer is $25$ .

$25$

Step 4

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

$(5, 25)$

Step 5

Find the Maximum/Minimum Value m(x)=10x-x^2

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