Find the Maximum/Minimum Value r(p)=-2.5p^2+750p

$r (p) = - 2.5 p^{2} + 750 p$

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

$f_{max}$ $p = a p^{2} + b p + c$ occurs at $p = - \frac{b}{2 a}$

Find the value of $p = - \frac{b}{2 a}$ .

Substitute in the values of $a$ and $b$ .

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

Remove parentheses.

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

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

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

Factor $2$ out of $750$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

Divide $375$ by $- 2.5$ .

$p = - - 150$

Multiply $- 1$ by $- 150$ .

$p = 150$

Evaluate $f (150)$ .

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $- 2.5$ by $22500$ .

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

Multiply $750$ by $150$ .

$r (150) = - 56250 + 112500$

Add $- 56250$ and $112500$ .

$r (150) = 56250$

The final answer is $56250$ .

$56250$

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

$(150, 56250)$

Find the Maximum/Minimum Value r(p)=-2.5p^2+750p

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