Find the Maximum/Minimum Value f(x)=56x-16x^2

$f (x) = 56 x - 16 x^{2}$

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 $b$ .

$x = - \frac{56}{2 (- 16)}$

Remove parentheses.

$x = - \frac{56}{2 (- 16)}$

Simplify $- \frac{56}{2 (- 16)}$ .

Cancel the common factor of $56$ and $2$ .

Factor $2$ out of $56$ .

$x = - \frac{2 \cdot 28}{2 \cdot - 16}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 28}{2 (- 16)}$

Cancel the common factor.

$x = - \frac{2 \cdot 28}{2 \cdot - 16}$

Rewrite the expression.

$x = - \frac{28}{- 16}$

Cancel the common factor of $28$ and $- 16$ .

Factor $4$ out of $28$ .

$x = - \frac{4 (7)}{- 16}$

Cancel the common factors.

Factor $4$ out of $- 16$ .

$x = - \frac{4 \cdot 7}{4 \cdot - 4}$

Cancel the common factor.

$x = - \frac{4 \cdot 7}{4 \cdot - 4}$

Rewrite the expression.

$x = - \frac{7}{- 4}$

Move the negative in front of the fraction.

$x = - - \frac{7}{4}$

Multiply $- - \frac{7}{4}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{7}{4})$

Multiply $\frac{7}{4}$ by $1$ .

$x = \frac{7}{4}$

Step 3

Evaluate $f (\frac{7}{4})$ .

Replace the variable $x$ with $\frac{7}{4}$ in the expression.

$f (\frac{7}{4}) = 56 (\frac{7}{4}) - 16 {(\frac{7}{4})}^{2}$

Simplify the result.

Simplify each term.

Cancel the common factor of $4$ .

Factor $4$ out of $56$ .

$f (\frac{7}{4}) = 4 (14) (\frac{7}{4}) - 16 {(\frac{7}{4})}^{2}$

Cancel the common factor.

$f (\frac{7}{4}) = 4 \cdot (14 (\frac{7}{4})) - 16 {(\frac{7}{4})}^{2}$

Rewrite the expression.

$f (\frac{7}{4}) = 14 \cdot 7 - 16 {(\frac{7}{4})}^{2}$

Multiply $14$ by $7$ .

$f (\frac{7}{4}) = 98 - 16 {(\frac{7}{4})}^{2}$

Apply the product rule to $\frac{7}{4}$ .

$f (\frac{7}{4}) = 98 - 16 \frac{7^{2}}{4^{2}}$

Raise $7$ to the power of $2$ .

$f (\frac{7}{4}) = 98 - 16 \frac{49}{4^{2}}$

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

$f (\frac{7}{4}) = 98 - 16 (\frac{49}{16})$

Cancel the common factor of $16$ .

Factor $16$ out of $- 16$ .

$f (\frac{7}{4}) = 98 + 16 (- 1) (\frac{49}{16})$

Cancel the common factor.

$f (\frac{7}{4}) = 98 + 16 \cdot (- 1 (\frac{49}{16}))$

Rewrite the expression.

$f (\frac{7}{4}) = 98 - 1 \cdot 49$

Multiply $- 1$ by $49$ .

$f (\frac{7}{4}) = 98 - 49$

Subtract $49$ from $98$ .

$f (\frac{7}{4}) = 49$

The final answer is $49$ .

$49$

Step 4

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

$(\frac{7}{4}, 49)$

Step 5

Find the Maximum/Minimum Value f(x)=56x-16x^2

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