Find the Maximum/Minimum Value A=-2x^2+800x

$A = - 2 x^{2} + 800 x$

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 $b$ .

$x = - \frac{800}{2 (- 2)}$

Remove parentheses.

$x = - \frac{800}{2 (- 2)}$

Simplify $- \frac{800}{2 (- 2)}$ .

Cancel the common factor of $800$ and $2$ .

Factor $2$ out of $800$ .

$x = - \frac{2 \cdot 400}{2 \cdot - 2}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 400}{2 (- 2)}$

Cancel the common factor.

$x = - \frac{2 \cdot 400}{2 \cdot - 2}$

Rewrite the expression.

$x = - \frac{400}{- 2}$

Cancel the common factor of $400$ and $- 2$ .

Factor $2$ out of $400$ .

$x = - \frac{2 (200)}{- 2}$

Move the negative one from the denominator of $\frac{200}{- 1}$ .

$x = - (- 1 \cdot 200)$

Multiply $- (- 1 \cdot 200)$ .

Multiply $- 1$ by $200$ .

$x = - - 200$

Multiply $- 1$ by $- 200$ .

$x = 200$

Step 3

Evaluate $f (200)$ .

Replace the variable $x$ with $200$ in the expression.

$f (200) = - 2 {(200)}^{2} + 800 (200)$

Simplify the result.

Simplify each term.

Raise $200$ to the power of $2$ .

$f (200) = - 2 \cdot 40000 + 800 (200)$

Multiply $- 2$ by $40000$ .

$f (200) = - 80000 + 800 (200)$

Multiply $800$ by $200$ .

$f (200) = - 80000 + 160000$

Add $- 80000$ and $160000$ .

$f (200) = 80000$

The final answer is $80000$ .

$80000$

Step 4

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

$(200, 80000)$

Step 5

Find the Maximum/Minimum Value A=-2x^2+800x

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