Find the Maximum/Minimum Value f(p)=-50p^2+1700p-12000

$f (p) = - 50 p^{2} + 1700 p - 12000$

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 $f (- \frac{b}{2 a})$ .

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

Step 2

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

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

$p = - \frac{1700}{2 (- 50)}$

Remove parentheses.

$p = - \frac{1700}{2 (- 50)}$

Simplify $- \frac{1700}{2 (- 50)}$ .

Cancel the common factor of $1700$ and $2$ .

Factor $2$ out of $1700$ .

$p = - \frac{2 \cdot 850}{2 \cdot - 50}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 50$ .

$p = - \frac{2 \cdot 850}{2 (- 50)}$

Cancel the common factor.

$p = - \frac{2 \cdot 850}{2 \cdot - 50}$

Rewrite the expression.

$p = - \frac{850}{- 50}$

Cancel the common factor of $850$ and $- 50$ .

Factor $50$ out of $850$ .

$p = - \frac{50 (17)}{- 50}$

Move the negative one from the denominator of $\frac{17}{- 1}$ .

$p = - (- 1 \cdot 17)$

Multiply $- (- 1 \cdot 17)$ .

Multiply $- 1$ by $17$ .

$p = - - 17$

Multiply $- 1$ by $- 17$ .

$p = 17$

Step 3

Evaluate $f (17)$ .

Replace the variable $p$ with $17$ in the expression.

$f (17) = - 50 {(17)}^{2} + 1700 (17) - 12000$

Simplify the result.

Simplify each term.

Raise $17$ to the power of $2$ .

$f (17) = - 50 \cdot 289 + 1700 (17) - 12000$

Multiply $- 50$ by $289$ .

$f (17) = - 14450 + 1700 (17) - 12000$

Multiply $1700$ by $17$ .

$f (17) = - 14450 + 28900 - 12000$

Simplify by adding and subtracting.

Add $- 14450$ and $28900$ .

$f (17) = 14450 - 12000$

Subtract $12000$ from $14450$ .

$f (17) = 2450$

The final answer is $2450$ .

$2450$

Step 4

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

$(17, 2450)$

Step 5

Find the Maximum/Minimum Value f(p)=-50p^2+1700p-12000

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