Find the Maximum/Minimum Value f(x)=2x^2-14x+4

$f (x) = 2 x^{2} - 14 x + 4$

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 $b$ .

$x = - \frac{- 14}{2 (2)}$

Remove parentheses.

$x = - \frac{- 14}{2 (2)}$

Simplify $- \frac{- 14}{2 (2)}$ .

Cancel the common factor of $- 14$ and $2$ .

Factor $2$ out of $- 14$ .

$x = - \frac{2 \cdot - 7}{2 \cdot 2}$

Cancel the common factors.

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

$x = - \frac{2 \cdot - 7}{2 (2)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 7}{2 \cdot 2}$

Rewrite the expression.

$x = - \frac{- 7}{2}$

Move the negative in front of the fraction.

$x = - - \frac{7}{2}$

Multiply $- - \frac{7}{2}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{7}{2})$

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

$x = \frac{7}{2}$

Step 3

Evaluate $f (\frac{7}{2})$ .

Replace the variable $x$ with $\frac{7}{2}$ in the expression.

$f (\frac{7}{2}) = 2 {(\frac{7}{2})}^{2} - 14 (\frac{7}{2}) + 4$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{7}{2}$ .

$f (\frac{7}{2}) = 2 (\frac{7^{2}}{2^{2}}) - 14 (\frac{7}{2}) + 4$

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

$f (\frac{7}{2}) = 2 (\frac{49}{2^{2}}) - 14 (\frac{7}{2}) + 4$

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

$f (\frac{7}{2}) = 2 (\frac{49}{4}) - 14 (\frac{7}{2}) + 4$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$f (\frac{7}{2}) = 2 (\frac{49}{2 (2)}) - 14 (\frac{7}{2}) + 4$

Cancel the common factor.

$f (\frac{7}{2}) = 2 (\frac{49}{2 \cdot 2}) - 14 (\frac{7}{2}) + 4$

Rewrite the expression.

$f (\frac{7}{2}) = \frac{49}{2} - 14 (\frac{7}{2}) + 4$

Cancel the common factor of $2$ .

Factor $2$ out of $- 14$ .

$f (\frac{7}{2}) = \frac{49}{2} + 2 (- 7) (\frac{7}{2}) + 4$

Cancel the common factor.

$f (\frac{7}{2}) = \frac{49}{2} + 2 \cdot (- 7 (\frac{7}{2})) + 4$

Rewrite the expression.

$f (\frac{7}{2}) = \frac{49}{2} - 7 \cdot 7 + 4$

Multiply $- 7$ by $7$ .

$f (\frac{7}{2}) = \frac{49}{2} - 49 + 4$

Find the common denominator.

Write $- 49$ as a fraction with denominator $1$ .

$f (\frac{7}{2}) = \frac{49}{2} + \frac{- 49}{1} + 4$

Multiply $\frac{- 49}{1}$ by $\frac{2}{2}$ .

$f (\frac{7}{2}) = \frac{49}{2} + \frac{- 49}{1} \cdot \frac{2}{2} + 4$

Multiply $\frac{- 49}{1}$ by $\frac{2}{2}$ .

$f (\frac{7}{2}) = \frac{49}{2} + \frac{- 49 \cdot 2}{2} + 4$

Write $4$ as a fraction with denominator $1$ .

$f (\frac{7}{2}) = \frac{49}{2} + \frac{- 49 \cdot 2}{2} + \frac{4}{1}$

Multiply $\frac{4}{1}$ by $\frac{2}{2}$ .

$f (\frac{7}{2}) = \frac{49}{2} + \frac{- 49 \cdot 2}{2} + \frac{4}{1} \cdot \frac{2}{2}$

Multiply $\frac{4}{1}$ by $\frac{2}{2}$ .

$f (\frac{7}{2}) = \frac{49}{2} + \frac{- 49 \cdot 2}{2} + \frac{4 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (\frac{7}{2}) = \frac{49 - 49 \cdot 2 + 4 \cdot 2}{2}$

Simplify each term.

Multiply $- 49$ by $2$ .

$f (\frac{7}{2}) = \frac{49 - 98 + 4 \cdot 2}{2}$

Multiply $4$ by $2$ .

$f (\frac{7}{2}) = \frac{49 - 98 + 8}{2}$

Simplify the expression.

Subtract $98$ from $49$ .

$f (\frac{7}{2}) = \frac{- 49 + 8}{2}$

Add $- 49$ and $8$ .

$f (\frac{7}{2}) = \frac{- 41}{2}$

Move the negative in front of the fraction.

$f (\frac{7}{2}) = - \frac{41}{2}$

The final answer is $- \frac{41}{2}$ .

$- \frac{41}{2}$

Step 4

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

$(\frac{7}{2}, - \frac{41}{2})$

Step 5

Find the Maximum/Minimum Value f(x)=2x^2-14x+4

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