Find the Maximum/Minimum Value f(x)=3(x-2)^2+7

$f (x) = 3 {(x - 2)}^{2} + 7$

Step 1

Simplify each term.

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

$f (x) = 3 ((x - 2) (x - 2)) + 7$

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$f (x) = 3 (x (x - 2) - 2 (x - 2)) + 7$

Apply the distributive property.

$f (x) = 3 (x \cdot x + x \cdot - 2 - 2 (x - 2)) + 7$

Apply the distributive property.

$f (x) = 3 (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) + 7$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = 3 (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) + 7$

Move $- 2$ to the left of $x$ .

$f (x) = 3 (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) + 7$

Multiply $- 2$ by $- 2$ .

$f (x) = 3 (x^{2} - 2 x - 2 x + 4) + 7$

Subtract $2 x$ from $- 2 x$ .

$f (x) = 3 (x^{2} - 4 x + 4) + 7$

Apply the distributive property.

$f (x) = 3 x^{2} + 3 (- 4 x) + 3 \cdot 4 + 7$

Simplify.

Multiply $- 4$ by $3$ .

$f (x) = 3 x^{2} - 12 x + 3 \cdot 4 + 7$

Multiply $3$ by $4$ .

$f (x) = 3 x^{2} - 12 x + 12 + 7$

Step 2

Add $12$ and $7$ .

$f (x) = 3 x^{2} - 12 x + 19$

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 $b$ .

$x = - \frac{- 12}{2 (3)}$

Remove parentheses.

$x = - \frac{- 12}{2 (3)}$

Simplify $- \frac{- 12}{2 (3)}$ .

Cancel the common factor of $- 12$ and $2$ .

Factor $2$ out of $- 12$ .

$x = - \frac{2 \cdot - 6}{2 \cdot 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 3$ .

$x = - \frac{2 \cdot - 6}{2 (3)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 6}{2 \cdot 3}$

Rewrite the expression.

$x = - \frac{- 6}{3}$

Cancel the common factor of $- 6$ and $3$ .

Factor $3$ out of $- 6$ .

$x = - \frac{3 \cdot - 2}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$x = - \frac{3 \cdot - 2}{3 (1)}$

Cancel the common factor.

$x = - \frac{3 \cdot - 2}{3 \cdot 1}$

Rewrite the expression.

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

Divide $- 2$ by $1$ .

$x = - - 2$

Multiply $- 1$ by $- 2$ .

$x = 2$

Step 5

Evaluate $f (2)$ .

Replace the variable $x$ with $2$ in the expression.

$f (2) = 3 {(2)}^{2} - 12 \cdot 2 + 19$

Simplify the result.

Simplify each term.

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

$f (2) = 3 \cdot 4 - 12 \cdot 2 + 19$

Multiply $3$ by $4$ .

$f (2) = 12 - 12 \cdot 2 + 19$

Multiply $- 12$ by $2$ .

$f (2) = 12 - 24 + 19$

Simplify by adding and subtracting.

Subtract $24$ from $12$ .

$f (2) = - 12 + 19$

Add $- 12$ and $19$ .

$f (2) = 7$

The final answer is $7$ .

$7$

Step 6

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

$(2, 7)$

Step 7

Find the Maximum/Minimum Value f(x)=3(x-2)^2+7

Related Questions

Support LCMGCF.com by disabling your adblocker.