Find the Maximum/Minimum Value R(x)=300x-0.5x^2

$R (x) = 300 x - 0.5 x^{2}$

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 $b$ .

$x = - \frac{300}{2 (- 0.5)}$

Remove parentheses.

$x = - \frac{300}{2 (- 0.5)}$

Simplify $- \frac{300}{2 (- 0.5)}$ .

Cancel the common factor of $300$ and $2$ .

Factor $2$ out of $300$ .

$x = - \frac{2 \cdot 150}{2 \cdot - 0.5}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 150}{2 (- 0.5)}$

Cancel the common factor.

$x = - \frac{2 \cdot 150}{2 \cdot - 0.5}$

Rewrite the expression.

$x = - \frac{150}{- 0.5}$

Simplify the expression.

Divide $150$ by $- 0.5$ .

$x = - - 300$

Multiply $- 1$ by $- 300$ .

$x = 300$

Step 3

Evaluate $f (300)$ .

Replace the variable $x$ with $300$ in the expression.

$R (300) = 300 (300) - 0.5 {(300)}^{2}$

Simplify the result.

Simplify each term.

Multiply $300$ by $300$ .

$R (300) = 90000 - 0.5 {(300)}^{2}$

Raise $300$ to the power of $2$ .

$R (300) = 90000 - 0.5 \cdot 90000$

Multiply $- 0.5$ by $90000$ .

$R (300) = 90000 - 45000$

Subtract $45000$ from $90000$ .

$R (300) = 45000$

The final answer is $45000$ .

$45000$

Step 4

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

$(300, 45000)$

Step 5

Find the Maximum/Minimum Value R(x)=300x-0.5x^2

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