Find the Maximum/Minimum Value f(x)=5x^2-10x+10

$f (x) = 5 x^{2} - 10 x + 10$

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 $b$ .

$x = - \frac{- 10}{2 (5)}$

Remove parentheses.

$x = - \frac{- 10}{2 (5)}$

Simplify $- \frac{- 10}{2 (5)}$ .

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10$ .

$x = - \frac{2 \cdot - 5}{2 \cdot 5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 5$ .

$x = - \frac{2 \cdot - 5}{2 (5)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 5}{2 \cdot 5}$

Rewrite the expression.

$x = - \frac{- 5}{5}$

Cancel the common factor of $- 5$ and $5$ .

Factor $5$ out of $- 5$ .

$x = - \frac{5 \cdot - 1}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$x = - \frac{5 \cdot - 1}{5 (1)}$

Cancel the common factor.

$x = - \frac{5 \cdot - 1}{5 \cdot 1}$

Rewrite the expression.

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

Divide $- 1$ by $1$ .

$x = - - 1$

Multiply $- 1$ by $- 1$ .

$x = 1$

Step 3

Evaluate $f (1)$ .

Replace the variable $x$ with $1$ in the expression.

$f (1) = 5 {(1)}^{2} - 10 \cdot 1 + 10$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 5 \cdot 1 - 10 \cdot 1 + 10$

Multiply $5$ by $1$ .

$f (1) = 5 - 10 \cdot 1 + 10$

Multiply $- 10$ by $1$ .

$f (1) = 5 - 10 + 10$

Simplify by adding and subtracting.

Subtract $10$ from $5$ .

$f (1) = - 5 + 10$

Add $- 5$ and $10$ .

$f (1) = 5$

The final answer is $5$ .

$5$

Step 4

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

$(1, 5)$

Step 5

Find the Maximum/Minimum Value f(x)=5x^2-10x+10

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