Find the Maximum/Minimum Value f(x)=2x^2-10x+14

$f (x) = 2 x^{2} - 10 x + 14$

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 $b$ .

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

Remove parentheses.

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

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

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

Factor $2$ out of $- 10$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Multiply $- - \frac{5}{2}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{5}{2})$

Multiply $\frac{5}{2}$ by $1$ .

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

Step 3

Evaluate $f (\frac{5}{2})$ .

Replace the variable $x$ with $\frac{5}{2}$ in the expression.

$f (\frac{5}{2}) = 2 {(\frac{5}{2})}^{2} - 10 (\frac{5}{2}) + 14$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{5}{2}$ .

$f (\frac{5}{2}) = 2 (\frac{5^{2}}{2^{2}}) - 10 (\frac{5}{2}) + 14$

Raise $5$ to the power of $2$ .

$f (\frac{5}{2}) = 2 (\frac{25}{2^{2}}) - 10 (\frac{5}{2}) + 14$

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

$f (\frac{5}{2}) = 2 (\frac{25}{4}) - 10 (\frac{5}{2}) + 14$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$f (\frac{5}{2}) = 2 (\frac{25}{2 (2)}) - 10 (\frac{5}{2}) + 14$

Cancel the common factor.

$f (\frac{5}{2}) = 2 (\frac{25}{2 \cdot 2}) - 10 (\frac{5}{2}) + 14$

Rewrite the expression.

$f (\frac{5}{2}) = \frac{25}{2} - 10 (\frac{5}{2}) + 14$

Cancel the common factor of $2$ .

Factor $2$ out of $- 10$ .

$f (\frac{5}{2}) = \frac{25}{2} + 2 (- 5) (\frac{5}{2}) + 14$

Cancel the common factor.

$f (\frac{5}{2}) = \frac{25}{2} + 2 \cdot (- 5 (\frac{5}{2})) + 14$

Rewrite the expression.

$f (\frac{5}{2}) = \frac{25}{2} - 5 \cdot 5 + 14$

Multiply $- 5$ by $5$ .

$f (\frac{5}{2}) = \frac{25}{2} - 25 + 14$

Find the common denominator.

Write $- 25$ as a fraction with denominator $1$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25}{1} + 14$

Multiply $\frac{- 25}{1}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25}{1} \cdot \frac{2}{2} + 14$

Multiply $\frac{- 25}{1}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25 \cdot 2}{2} + 14$

Write $14$ as a fraction with denominator $1$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25 \cdot 2}{2} + \frac{14}{1}$

Multiply $\frac{14}{1}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25 \cdot 2}{2} + \frac{14}{1} \cdot \frac{2}{2}$

Multiply $\frac{14}{1}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = \frac{25}{2} + \frac{- 25 \cdot 2}{2} + \frac{14 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (\frac{5}{2}) = \frac{25 - 25 \cdot 2 + 14 \cdot 2}{2}$

Simplify each term.

Multiply $- 25$ by $2$ .

$f (\frac{5}{2}) = \frac{25 - 50 + 14 \cdot 2}{2}$

Multiply $14$ by $2$ .

$f (\frac{5}{2}) = \frac{25 - 50 + 28}{2}$

Simplify by adding and subtracting.

Subtract $50$ from $25$ .

$f (\frac{5}{2}) = \frac{- 25 + 28}{2}$

Add $- 25$ and $28$ .

$f (\frac{5}{2}) = \frac{3}{2}$

The final answer is $\frac{3}{2}$ .

$\frac{3}{2}$

Step 4

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

$(\frac{5}{2}, \frac{3}{2})$

Step 5

Find the Maximum/Minimum Value f(x)=2x^2-10x+14

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