Find the Maximum/Minimum Value 104x-16x^2

$104 x - 16 x^{2}$

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 $b$ .

$x = - \frac{104}{2 (- 16)}$

Remove parentheses.

$x = - \frac{104}{2 (- 16)}$

Simplify $- \frac{104}{2 (- 16)}$ .

Cancel the common factor of $104$ and $2$ .

Factor $2$ out of $104$ .

$x = - \frac{2 \cdot 52}{2 \cdot - 16}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 52}{2 (- 16)}$

Cancel the common factor.

$x = - \frac{2 \cdot 52}{2 \cdot - 16}$

Rewrite the expression.

$x = - \frac{52}{- 16}$

Cancel the common factor of $52$ and $- 16$ .

Factor $4$ out of $52$ .

$x = - \frac{4 (13)}{- 16}$

Cancel the common factors.

Factor $4$ out of $- 16$ .

$x = - \frac{4 \cdot 13}{4 \cdot - 4}$

Cancel the common factor.

$x = - \frac{4 \cdot 13}{4 \cdot - 4}$

Rewrite the expression.

$x = - \frac{13}{- 4}$

Move the negative in front of the fraction.

$x = - - \frac{13}{4}$

Multiply $- - \frac{13}{4}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{13}{4})$

Multiply $\frac{13}{4}$ by $1$ .

$x = \frac{13}{4}$

Step 3

Evaluate $f (\frac{13}{4})$ .

Replace the variable $x$ with $\frac{13}{4}$ in the expression.

$f (\frac{13}{4}) = 104 (\frac{13}{4}) - 16 {(\frac{13}{4})}^{2}$

Simplify the result.

Simplify each term.

Cancel the common factor of $4$ .

Factor $4$ out of $104$ .

$f (\frac{13}{4}) = 4 (26) (\frac{13}{4}) - 16 {(\frac{13}{4})}^{2}$

Cancel the common factor.

$f (\frac{13}{4}) = 4 \cdot (26 (\frac{13}{4})) - 16 {(\frac{13}{4})}^{2}$

Rewrite the expression.

$f (\frac{13}{4}) = 26 \cdot 13 - 16 {(\frac{13}{4})}^{2}$

Multiply $26$ by $13$ .

$f (\frac{13}{4}) = 338 - 16 {(\frac{13}{4})}^{2}$

Apply the product rule to $\frac{13}{4}$ .

$f (\frac{13}{4}) = 338 - 16 \frac{13^{2}}{4^{2}}$

Raise $13$ to the power of $2$ .

$f (\frac{13}{4}) = 338 - 16 \frac{169}{4^{2}}$

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

$f (\frac{13}{4}) = 338 - 16 (\frac{169}{16})$

Cancel the common factor of $16$ .

Factor $16$ out of $- 16$ .

$f (\frac{13}{4}) = 338 + 16 (- 1) (\frac{169}{16})$

Cancel the common factor.

$f (\frac{13}{4}) = 338 + 16 \cdot (- 1 (\frac{169}{16}))$

Rewrite the expression.

$f (\frac{13}{4}) = 338 - 1 \cdot 169$

Multiply $- 1$ by $169$ .

$f (\frac{13}{4}) = 338 - 169$

Subtract $169$ from $338$ .

$f (\frac{13}{4}) = 169$

The final answer is $169$ .

$169$

Step 4

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

$(\frac{13}{4}, 169)$

Step 5

Find the Maximum/Minimum Value 104x-16x^2

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