Find the Maximum/Minimum Value f(x)=2(x-4)^2-6

$f (x) = 2 {(x - 4)}^{2} - 6$

Step 1

Simplify each term.

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$f (x) = 2 ((x - 4) (x - 4)) - 6$

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$f (x) = 2 (x (x - 4) - 4 (x - 4)) - 6$

Apply the distributive property.

$f (x) = 2 (x \cdot x + x \cdot - 4 - 4 (x - 4)) - 6$

Apply the distributive property.

$f (x) = 2 (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) - 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = 2 (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4) - 6$

Move $- 4$ to the left of $x$ .

$f (x) = 2 (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) - 6$

Multiply $- 4$ by $- 4$ .

$f (x) = 2 (x^{2} - 4 x - 4 x + 16) - 6$

Subtract $4 x$ from $- 4 x$ .

$f (x) = 2 (x^{2} - 8 x + 16) - 6$

Apply the distributive property.

$f (x) = 2 x^{2} + 2 (- 8 x) + 2 \cdot 16 - 6$

Simplify.

Multiply $- 8$ by $2$ .

$f (x) = 2 x^{2} - 16 x + 2 \cdot 16 - 6$

Multiply $2$ by $16$ .

$f (x) = 2 x^{2} - 16 x + 32 - 6$

Step 2

Subtract $6$ from $32$ .

$f (x) = 2 x^{2} - 16 x + 26$

Step 3

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 4

Find the value of $x = - \frac{b}{2 a}$ .

Substitute in the values of $a$ and $b$ .

$x = - \frac{- 16}{2 (2)}$

Remove parentheses.

$x = - \frac{- 16}{2 (2)}$

Simplify $- \frac{- 16}{2 (2)}$ .

Cancel the common factor of $- 16$ and $2$ .

Factor $2$ out of $- 16$ .

$x = - \frac{2 \cdot - 8}{2 \cdot 2}$

Cancel the common factors.

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

$x = - \frac{2 \cdot - 8}{2 (2)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 8}{2 \cdot 2}$

Rewrite the expression.

$x = - \frac{- 8}{2}$

Cancel the common factor of $- 8$ and $2$ .

Factor $2$ out of $- 8$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

$x = - \frac{2 \cdot - 4}{2 \cdot 1}$

Rewrite the expression.

$x = - \frac{- 4}{1}$

Divide $- 4$ by $1$ .

$x = - - 4$

Multiply $- 1$ by $- 4$ .

$x = 4$

Step 5

Evaluate $f (4)$ .

Replace the variable $x$ with $4$ in the expression.

$f (4) = 2 {(4)}^{2} - 16 \cdot 4 + 26$

Simplify the result.

Simplify each term.

Raise $4$ to the power of $2$ .

$f (4) = 2 \cdot 16 - 16 \cdot 4 + 26$

Multiply $2$ by $16$ .

$f (4) = 32 - 16 \cdot 4 + 26$

Multiply $- 16$ by $4$ .

$f (4) = 32 - 64 + 26$

Simplify by adding and subtracting.

Subtract $64$ from $32$ .

$f (4) = - 32 + 26$

Add $- 32$ and $26$ .

$f (4) = - 6$

The final answer is $- 6$ .

$- 6$

Step 6

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

$(4, - 6)$

Step 7

Find the Maximum/Minimum Value f(x)=2(x-4)^2-6

Related Questions

Support LCMGCF.com by disabling your adblocker.