Find the Maximum/Minimum Value -4x^2+4x-2

$- 4 x^{2} + 4 x - 2$

Step 1

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

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

$x = - \frac{4}{2 (- 4)}$

Remove parentheses.

$x = - \frac{4}{2 (- 4)}$

Simplify $- \frac{4}{2 (- 4)}$ .

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$x = - \frac{2 \cdot 2}{2 \cdot - 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 4$ .

$x = - \frac{2 \cdot 2}{2 (- 4)}$

Cancel the common factor.

$x = - \frac{2 \cdot 2}{2 \cdot - 4}$

Rewrite the expression.

$x = - \frac{2}{- 4}$

Cancel the common factor of $2$ and $- 4$ .

Factor $2$ out of $2$ .

$x = - \frac{2 (1)}{- 4}$

Cancel the common factors.

Factor $2$ out of $- 4$ .

$x = - \frac{2 \cdot 1}{2 \cdot - 2}$

Cancel the common factor.

$x = - \frac{2 \cdot 1}{2 \cdot - 2}$

Rewrite the expression.

$x = - \frac{1}{- 2}$

Move the negative in front of the fraction.

$x = - - \frac{1}{2}$

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

Multiply $- 1$ by $- 1$ .

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

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

$x = \frac{1}{2}$

Step 3

Evaluate $f (\frac{1}{2})$ .

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

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

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{1}{2}$ .

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

One to any power is one.

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

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

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

Cancel the common factor of $4$ .

Factor $4$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify by adding and subtracting.

Add $- 1$ and $2$ .

$f (\frac{1}{2}) = 1 - 2$

Subtract $2$ from $1$ .

$f (\frac{1}{2}) = - 1$

The final answer is $- 1$ .

$- 1$

Step 4

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

$(\frac{1}{2}, - 1)$

Step 5

Find the Maximum/Minimum Value -4x^2+4x-2

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