Find the Maximum/Minimum Value f(x)=-1/2x^2+7x-4

$f (x) = - \frac{1}{2} x^{2} + 7 x - 4$

Step 1

Combine $x^{2}$ and $\frac{1}{2}$ .

$f (x) = - \frac{x^{2}}{2} + 7 x - 4$

Step 2

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 3

Find the value of $x = - \frac{b}{2 a}$ .

Substitute in the values of $a$ and $b$ .

$x = - \frac{7}{2 (- 0.5)}$

Remove parentheses.

$x = - \frac{7}{2 (- 0.5)}$

Simplify $- \frac{7}{2 (- 0.5)}$ .

Multiply $2$ by $- 0.5$ .

$x = - \frac{7}{- 1}$

Divide $7$ by $- 1$ .

$x = - - 7$

Multiply $- 1$ by $- 7$ .

$x = 7$

Step 4

Evaluate $f (7)$ .

Replace the variable $x$ with $7$ in the expression.

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

Simplify the result.

Simplify each term.

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

$f (7) = - \frac{49}{2} + 7 (7) - 4$

Multiply $7$ by $7$ .

$f (7) = - \frac{49}{2} + 49 - 4$

Find the common denominator.

Write $49$ as a fraction with denominator $1$ .

$f (7) = - \frac{49}{2} + \frac{49}{1} - 4$

Multiply $\frac{49}{1}$ by $\frac{2}{2}$ .

$f (7) = - \frac{49}{2} + \frac{49}{1} \cdot \frac{2}{2} - 4$

Multiply $\frac{49}{1}$ by $\frac{2}{2}$ .

$f (7) = - \frac{49}{2} + \frac{49 \cdot 2}{2} - 4$

Write $- 4$ as a fraction with denominator $1$ .

$f (7) = - \frac{49}{2} + \frac{49 \cdot 2}{2} + \frac{- 4}{1}$

Multiply $\frac{- 4}{1}$ by $\frac{2}{2}$ .

$f (7) = - \frac{49}{2} + \frac{49 \cdot 2}{2} + \frac{- 4}{1} \cdot \frac{2}{2}$

Multiply $\frac{- 4}{1}$ by $\frac{2}{2}$ .

$f (7) = - \frac{49}{2} + \frac{49 \cdot 2}{2} + \frac{- 4 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (7) = \frac{- 49 + 49 \cdot 2 - 4 \cdot 2}{2}$

Simplify each term.

Multiply $49$ by $2$ .

$f (7) = \frac{- 49 + 98 - 4 \cdot 2}{2}$

Multiply $- 4$ by $2$ .

$f (7) = \frac{- 49 + 98 - 8}{2}$

Simplify by adding and subtracting.

Add $- 49$ and $98$ .

$f (7) = \frac{49 - 8}{2}$

Subtract $8$ from $49$ .

$f (7) = \frac{41}{2}$

The final answer is $\frac{41}{2}$ .

$\frac{41}{2}$

Step 5

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

$(7, \frac{41}{2})$

Step 6

Find the Maximum/Minimum Value f(x)=-1/2x^2+7x-4

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