Find the Maximum/Minimum Value 1/2x^2-7x+7

$\frac{1}{2} x^{2} - 7 x + 7$

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

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

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 $x = - \frac{b}{2 a}$

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 $- 7$ .

$x = 7$

Evaluate $f (7)$ .

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

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

Simplify the result.

Find the common denominator.

Write $- 7 \cdot 7$ as a fraction with denominator $1$ .

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

Multiply $\frac{- 7 \cdot 7}{1}$ by $\frac{2}{2}$ .

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

Multiply $\frac{- 7 \cdot 7}{1}$ by $\frac{2}{2}$ .

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

Write $7$ as a fraction with denominator $1$ .

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

Multiply $\frac{7}{1}$ by $\frac{2}{2}$ .

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

Multiply $\frac{7}{1}$ by $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

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

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

Multiply $- 7$ by $7$ .

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

Multiply $- 49$ by $2$ .

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

Multiply $7$ by $2$ .

$f (7) = \frac{49 - 98 + 14}{2}$

Simplify the expression.

Subtract $98$ from $49$ .

$f (7) = \frac{- 49 + 14}{2}$

Add $- 49$ and $14$ .

$f (7) = \frac{- 35}{2}$

Move the negative in front of the fraction.

$f (7) = - \frac{35}{2}$

The final answer is $- \frac{35}{2}$ .

$- \frac{35}{2}$

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

$(7, - \frac{35}{2})$

Find the Maximum/Minimum Value 1/2x^2-7x+7

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