Find the Maximum/Minimum Value 100x-0.5x^2

$100 x - 0.5 x^{2}$

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 $b$ .

$x = - \frac{100}{2 (- 0.5)}$

Remove parentheses.

$x = - \frac{100}{2 (- 0.5)}$

Simplify $- \frac{100}{2 (- 0.5)}$ .

Cancel the common factor of $100$ and $2$ .

Factor $2$ out of $100$ .

$x = - \frac{2 \cdot 50}{2 \cdot - 0.5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 0.5$ .

$x = - \frac{2 \cdot 50}{2 (- 0.5)}$

Cancel the common factor.

$x = - \frac{2 \cdot 50}{2 \cdot - 0.5}$

Rewrite the expression.

$x = - \frac{50}{- 0.5}$

Simplify the expression.

Divide $50$ by $- 0.5$ .

$x = - - 100$

Multiply $- 1$ by $- 100$ .

$x = 100$

Step 3

Evaluate $f (100)$ .

Replace the variable $x$ with $100$ in the expression.

$f (100) = 100 (100) - 0.5 {(100)}^{2}$

Simplify the result.

Simplify each term.

Multiply $100$ by $100$ .

$f (100) = 10000 - 0.5 {(100)}^{2}$

Raise $100$ to the power of $2$ .

$f (100) = 10000 - 0.5 \cdot 10000$

Multiply $- 0.5$ by $10000$ .

$f (100) = 10000 - 5000$

Subtract $5000$ from $10000$ .

$f (100) = 5000$

The final answer is $5000$ .

$5000$

Step 4

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

$(100, 5000)$

Step 5

Find the Maximum/Minimum Value 100x-0.5x^2

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