Find the Maximum/Minimum Value p(x)=202x-5000-x^2

$p (x) = 202 x - 5000 - x^{2}$

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 $b$ .

$x = - \frac{202}{2 (- 1)}$

Remove parentheses.

$x = - \frac{202}{2 (- 1)}$

Simplify $- \frac{202}{2 (- 1)}$ .

Cancel the common factor of $202$ and $2$ .

Factor $2$ out of $202$ .

$x = - \frac{2 \cdot 101}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{101}{- 1}$ .

$x = - (- 1 \cdot 101)$

Multiply $- (- 1 \cdot 101)$ .

Multiply $- 1$ by $101$ .

$x = - - 101$

Multiply $- 1$ by $- 101$ .

$x = 101$

Step 3

Evaluate $f (101)$ .

Replace the variable $x$ with $101$ in the expression.

$p (101) = 202 (101) - 5000 - {(101)}^{2}$

Simplify the result.

Simplify each term.

Multiply $202$ by $101$ .

$p (101) = 20402 - 5000 - {(101)}^{2}$

Raise $101$ to the power of $2$ .

$p (101) = 20402 - 5000 - 1 \cdot 10201$

Multiply $- 1$ by $10201$ .

$p (101) = 20402 - 5000 - 10201$

Simplify by subtracting numbers.

Subtract $5000$ from $20402$ .

$p (101) = 15402 - 10201$

Subtract $10201$ from $15402$ .

$p (101) = 5201$

The final answer is $5201$ .

$5201$

Step 4

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

$(101, 5201)$

Step 5

Find the Maximum/Minimum Value p(x)=202x-5000-x^2

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