Find the Maximum/Minimum Value f(x)=(x^2)/2+6x+21

$f (x) = \frac{x^{2}}{2} + 6 x + 21$

Step 1

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{6}{2 (0.5)}$

Remove parentheses.

$x = - \frac{6}{2 (0.5)}$

Simplify $- \frac{6}{2 (0.5)}$ .

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

$x = - \frac{2 \cdot 3}{2 \cdot 0.5}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 3}{2 (0.5)}$

Cancel the common factor.

$x = - \frac{2 \cdot 3}{2 \cdot 0.5}$

Rewrite the expression.

$x = - \frac{3}{0.5}$

Simplify the expression.

Divide $3$ by $0.5$ .

$x = - 1 \cdot 6$

Multiply $- 1$ by $6$ .

$x = - 6$

Step 3

Evaluate $f (- 6)$ .

Replace the variable $x$ with $- 6$ in the expression.

$f (- 6) = \frac{{(- 6)}^{2}}{2} + 6 (- 6) + 21$

Simplify the result.

Simplify each term.

Raise $- 6$ to the power of $2$ .

$f (- 6) = \frac{36}{2} + 6 (- 6) + 21$

Divide $36$ by $2$ .

$f (- 6) = 18 + 6 (- 6) + 21$

Multiply $6$ by $- 6$ .

$f (- 6) = 18 - 36 + 21$

Simplify by adding and subtracting.

Subtract $36$ from $18$ .

$f (- 6) = - 18 + 21$

Add $- 18$ and $21$ .

$f (- 6) = 3$

The final answer is $3$ .

$3$

Step 4

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

$(- 6, 3)$

Step 5

Find the Maximum/Minimum Value f(x)=(x^2)/2+6x+21

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