Find the Maximum/Minimum Value f(x)=-0.1x^2+90x

$f (x) = - 0.1 x^{2} + 90 x$

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 $b$ .

$x = - \frac{90}{2 (- 0.1)}$

Remove parentheses.

$x = - \frac{90}{2 (- 0.1)}$

Simplify $- \frac{90}{2 (- 0.1)}$ .

Cancel the common factor of $90$ and $2$ .

Factor $2$ out of $90$ .

$x = - \frac{2 \cdot 45}{2 \cdot - 0.1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 0.1$ .

$x = - \frac{2 \cdot 45}{2 (- 0.1)}$

Cancel the common factor.

$x = - \frac{2 \cdot 45}{2 \cdot - 0.1}$

Rewrite the expression.

$x = - \frac{45}{- 0.1}$

Simplify the expression.

Divide $45$ by $- 0.1$ .

$x = - - 450$

Multiply $- 1$ by $- 450$ .

$x = 450$

Step 3

Evaluate $f (450)$ .

Replace the variable $x$ with $450$ in the expression.

$f (450) = - 0.1 {(450)}^{2} + 90 (450)$

Simplify the result.

Simplify each term.

Raise $450$ to the power of $2$ .

$f (450) = - 0.1 \cdot 202500 + 90 (450)$

Multiply $- 0.1$ by $202500$ .

$f (450) = - 20250 + 90 (450)$

Multiply $90$ by $450$ .

$f (450) = - 20250 + 40500$

Add $- 20250$ and $40500$ .

$f (450) = 20250$

The final answer is $20250$ .

$20250$

Step 4

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

$(450, 20250)$

Step 5

Find the Maximum/Minimum Value f(x)=-0.1x^2+90x

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