Find the Maximum/Minimum Value f(x)=-2.5x^2+750x

$f (x) = - 2.5 x^{2} + 750 x$

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 $b$ .

$x = - \frac{750}{2 (- 2.5)}$

Remove parentheses.

$x = - \frac{750}{2 (- 2.5)}$

Simplify $- \frac{750}{2 (- 2.5)}$ .

Cancel the common factor of $750$ and $2$ .

Factor $2$ out of $750$ .

$x = - \frac{2 \cdot 375}{2 \cdot - 2.5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 2.5$ .

$x = - \frac{2 \cdot 375}{2 (- 2.5)}$

Cancel the common factor.

$x = - \frac{2 \cdot 375}{2 \cdot - 2.5}$

Rewrite the expression.

$x = - \frac{375}{- 2.5}$

Simplify the expression.

Divide $375$ by $- 2.5$ .

$x = - - 150$

Multiply $- 1$ by $- 150$ .

$x = 150$

Step 3

Evaluate $f (150)$ .

Replace the variable $x$ with $150$ in the expression.

$f (150) = - 2.5 {(150)}^{2} + 750 (150)$

Simplify the result.

Simplify each term.

Raise $150$ to the power of $2$ .

$f (150) = - 2.5 \cdot 22500 + 750 (150)$

Multiply $- 2.5$ by $22500$ .

$f (150) = - 56250 + 750 (150)$

Multiply $750$ by $150$ .

$f (150) = - 56250 + 112500$

Add $- 56250$ and $112500$ .

$f (150) = 56250$

The final answer is $56250$ .

$56250$

Step 4

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

$(150, 56250)$

Step 5

Find the Maximum/Minimum Value f(x)=-2.5x^2+750x

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