Find the Maximum/Minimum Value f(x)=-x^2+13x-34

$f (x) = - x^{2} + 13 x - 34$

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 $b$ .

$x = - \frac{13}{2 (- 1)}$

Remove parentheses.

$x = - \frac{13}{2 (- 1)}$

Simplify $- \frac{13}{2 (- 1)}$ .

Multiply $2$ by $- 1$ .

$x = - \frac{13}{- 2}$

Move the negative in front of the fraction.

$x = - - \frac{13}{2}$

Multiply $- - \frac{13}{2}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{13}{2})$

Multiply $\frac{13}{2}$ by $1$ .

$x = \frac{13}{2}$

Step 3

Evaluate $f (\frac{13}{2})$ .

Replace the variable $x$ with $\frac{13}{2}$ in the expression.

$f (\frac{13}{2}) = - {(\frac{13}{2})}^{2} + 13 (\frac{13}{2}) - 34$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{13}{2}$ .

$f (\frac{13}{2}) = - \frac{13^{2}}{2^{2}} + 13 (\frac{13}{2}) - 34$

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

$f (\frac{13}{2}) = - \frac{169}{2^{2}} + 13 (\frac{13}{2}) - 34$

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

$f (\frac{13}{2}) = - \frac{169}{4} + 13 (\frac{13}{2}) - 34$

Multiply $13 (\frac{13}{2})$ .

Combine $13$ and $\frac{13}{2}$ .

$f (\frac{13}{2}) = - \frac{169}{4} + \frac{13 \cdot 13}{2} - 34$

Multiply $13$ by $13$ .

$f (\frac{13}{2}) = - \frac{169}{4} + \frac{169}{2} - 34$

Find the common denominator.

Multiply $\frac{169}{2}$ by $\frac{2}{2}$ .

$f (\frac{13}{2}) = - \frac{169}{4} + \frac{169}{2} \cdot \frac{2}{2} - 34$

Multiply $\frac{169}{2}$ by $\frac{2}{2}$ .

$f (\frac{13}{2}) = - \frac{169}{4} + \frac{169 \cdot 2}{2 \cdot 2} - 34$

Write $- 34$ as a fraction with denominator $1$ .

$f (\frac{13}{2}) = - \frac{169}{4} + \frac{169 \cdot 2}{2 \cdot 2} + \frac{- 34}{1}$

Multiply $\frac{- 34}{1}$ by $\frac{4}{4}$ .

$f (\frac{13}{2}) = - \frac{169}{4} + \frac{169 \cdot 2}{2 \cdot 2} + \frac{- 34}{1} \cdot \frac{4}{4}$

Multiply $\frac{- 34}{1}$ by $\frac{4}{4}$ .

$f (\frac{13}{2}) = - \frac{169}{4} + \frac{169 \cdot 2}{2 \cdot 2} + \frac{- 34 \cdot 4}{4}$

Multiply $2$ by $2$ .

$f (\frac{13}{2}) = - \frac{169}{4} + \frac{169 \cdot 2}{4} + \frac{- 34 \cdot 4}{4}$

Combine the numerators over the common denominator.

$f (\frac{13}{2}) = \frac{- 169 + 169 \cdot 2 - 34 \cdot 4}{4}$

Simplify each term.

Multiply $169$ by $2$ .

$f (\frac{13}{2}) = \frac{- 169 + 338 - 34 \cdot 4}{4}$

Multiply $- 34$ by $4$ .

$f (\frac{13}{2}) = \frac{- 169 + 338 - 136}{4}$

Simplify by adding and subtracting.

Add $- 169$ and $338$ .

$f (\frac{13}{2}) = \frac{169 - 136}{4}$

Subtract $136$ from $169$ .

$f (\frac{13}{2}) = \frac{33}{4}$

The final answer is $\frac{33}{4}$ .

$\frac{33}{4}$

Step 4

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

$(\frac{13}{2}, \frac{33}{4})$

Step 5

Find the Maximum/Minimum Value f(x)=-x^2+13x-34

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