Find the Maximum/Minimum Value -1/2x^2+4x-11

$- \frac{1}{2} x^{2} + 4 x - 11$

Step 1

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

$f (x) = - \frac{x^{2}}{2} + 4 x - 11$

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 $b$ .

$x = - \frac{4}{2 (- 0.5)}$

Remove parentheses.

$x = - \frac{4}{2 (- 0.5)}$

Simplify $- \frac{4}{2 (- 0.5)}$ .

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$x = - \frac{2 \cdot 2}{2 \cdot - 0.5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 0.5$ .

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

Cancel the common factor.

$x = - \frac{2 \cdot 2}{2 \cdot - 0.5}$

Rewrite the expression.

$x = - \frac{2}{- 0.5}$

Simplify the expression.

Divide $2$ by $- 0.5$ .

$x = - - 4$

Multiply $- 1$ by $- 4$ .

$x = 4$

Step 4

Evaluate $f (4)$ .

Replace the variable $x$ with $4$ in the expression.

$f (4) = - \frac{{(4)}^{2}}{2} + 4 (4) - 11$

Simplify the result.

Simplify each term.

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

$f (4) = - \frac{16}{2} + 4 (4) - 11$

Divide $16$ by $2$ .

$f (4) = - 1 \cdot 8 + 4 (4) - 11$

Multiply $- 1$ by $8$ .

$f (4) = - 8 + 4 (4) - 11$

Multiply $4$ by $4$ .

$f (4) = - 8 + 16 - 11$

Simplify by adding and subtracting.

Add $- 8$ and $16$ .

$f (4) = 8 - 11$

Subtract $11$ from $8$ .

$f (4) = - 3$

The final answer is $- 3$ .

$- 3$

Step 5

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

$(4, - 3)$

Step 6

Find the Maximum/Minimum Value -1/2x^2+4x-11

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