Find the Maximum/Minimum Value f(x)=120x-16x^2

$f (x) = 120 x - 16 x^{2}$

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 $b$ .

$x = - \frac{120}{2 (- 16)}$

Remove parentheses.

$x = - \frac{120}{2 (- 16)}$

Simplify $- \frac{120}{2 (- 16)}$ .

Cancel the common factor of $120$ and $2$ .

Factor $2$ out of $120$ .

$x = - \frac{2 \cdot 60}{2 \cdot - 16}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 16$ .

$x = - \frac{2 \cdot 60}{2 (- 16)}$

Cancel the common factor.

$x = - \frac{2 \cdot 60}{2 \cdot - 16}$

Rewrite the expression.

$x = - \frac{60}{- 16}$

Cancel the common factor of $60$ and $- 16$ .

Factor $4$ out of $60$ .

$x = - \frac{4 (15)}{- 16}$

Cancel the common factors.

Factor $4$ out of $- 16$ .

$x = - \frac{4 \cdot 15}{4 \cdot - 4}$

Cancel the common factor.

$x = - \frac{4 \cdot 15}{4 \cdot - 4}$

Rewrite the expression.

$x = - \frac{15}{- 4}$

Move the negative in front of the fraction.

$x = - - \frac{15}{4}$

Multiply $- - \frac{15}{4}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{15}{4})$

Multiply $\frac{15}{4}$ by $1$ .

$x = \frac{15}{4}$

Step 3

Evaluate $f (\frac{15}{4})$ .

Replace the variable $x$ with $\frac{15}{4}$ in the expression.

$f (\frac{15}{4}) = 120 (\frac{15}{4}) - 16 {(\frac{15}{4})}^{2}$

Simplify the result.

Simplify each term.

Cancel the common factor of $4$ .

Factor $4$ out of $120$ .

$f (\frac{15}{4}) = 4 (30) (\frac{15}{4}) - 16 {(\frac{15}{4})}^{2}$

Cancel the common factor.

$f (\frac{15}{4}) = 4 \cdot (30 (\frac{15}{4})) - 16 {(\frac{15}{4})}^{2}$

Rewrite the expression.

$f (\frac{15}{4}) = 30 \cdot 15 - 16 {(\frac{15}{4})}^{2}$

Multiply $30$ by $15$ .

$f (\frac{15}{4}) = 450 - 16 {(\frac{15}{4})}^{2}$

Apply the product rule to $\frac{15}{4}$ .

$f (\frac{15}{4}) = 450 - 16 \frac{15^{2}}{4^{2}}$

Raise $15$ to the power of $2$ .

$f (\frac{15}{4}) = 450 - 16 \frac{225}{4^{2}}$

Raise $4$ to the power of $2$ .

$f (\frac{15}{4}) = 450 - 16 (\frac{225}{16})$

Cancel the common factor of $16$ .

Factor $16$ out of $- 16$ .

$f (\frac{15}{4}) = 450 + 16 (- 1) (\frac{225}{16})$

Cancel the common factor.

$f (\frac{15}{4}) = 450 + 16 \cdot (- 1 (\frac{225}{16}))$

Rewrite the expression.

$f (\frac{15}{4}) = 450 - 1 \cdot 225$

Multiply $- 1$ by $225$ .

$f (\frac{15}{4}) = 450 - 225$

Subtract $225$ from $450$ .

$f (\frac{15}{4}) = 225$

The final answer is $225$ .

$225$

Step 4

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

$(\frac{15}{4}, 225)$

Step 5

Find the Maximum/Minimum Value f(x)=120x-16x^2

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