Find the Maximum/Minimum Value y=200x-4.9x^2

$y = 200 x - 4.9 x^{2}$

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 $b$ .

$x = - \frac{200}{2 (- 4.9)}$

Remove parentheses.

$x = - \frac{200}{2 (- 4.9)}$

Simplify $- \frac{200}{2 (- 4.9)}$ .

Cancel the common factor of $200$ and $2$ .

Factor $2$ out of $200$ .

$x = - \frac{2 \cdot 100}{2 \cdot - 4.9}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 4.9$ .

$x = - \frac{2 \cdot 100}{2 (- 4.9)}$

Cancel the common factor.

$x = - \frac{2 \cdot 100}{2 \cdot - 4.9}$

Rewrite the expression.

$x = - \frac{100}{- 4.9}$

Simplify the expression.

Divide $100$ by $- 4.9$ .

$x = - - 20.40816326$

Multiply $- 1$ by $- 20.40816326$ .

$x = 20.40816326$

Step 3

Evaluate $f (20.40816326)$ .

Replace the variable $x$ with $20.40816326$ in the expression.

$f (20.40816326) = 200 (20.40816326) - 4.9 {(20.40816326)}^{2}$

Simplify the result.

Simplify each term.

Multiply $200$ by $20.40816326$ .

$f (20.40816326) = 4081.63265306 - 4.9 {(20.40816326)}^{2}$

Raise $20.40816326$ to the power of $2$ .

$f (20.40816326) = 4081.63265306 - 4.9 \cdot 416.49312786$

Multiply $- 4.9$ by $416.49312786$ .

$f (20.40816326) = 4081.63265306 - 2040.81632653$

Subtract $2040.81632653$ from $4081.63265306$ .

$f (20.40816326) = 2040.81632653$

The final answer is $2040.81632653$ .

$2040.81632653$

Step 4

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

$(20.40816326, 2040.81632653)$

Step 5

Find the Maximum/Minimum Value y=200x-4.9x^2

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