Find the Maximum/Minimum Value g(x)=-3x^2+12x-14

$g (x) = - 3 x^{2} + 12 x - 14$

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 $b$ .

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

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 $6$ .

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

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$x = - (- 1 \cdot 2)$

Multiply $- (- 1 \cdot 2)$ .

Multiply $- 1$ by $2$ .

$x = - - 2$

Multiply $- 1$ by $- 2$ .

$x = 2$

Step 3

Evaluate $f (2)$ .

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

$g (2) = - 3 {(2)}^{2} + 12 (2) - 14$

Simplify the result.

Simplify each term.

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

$g (2) = - 3 \cdot 4 + 12 (2) - 14$

Multiply $- 3$ by $4$ .

$g (2) = - 12 + 12 (2) - 14$

Multiply $12$ by $2$ .

$g (2) = - 12 + 24 - 14$

Simplify by adding and subtracting.

Add $- 12$ and $24$ .

$g (2) = 12 - 14$

Subtract $14$ from $12$ .

$g (2) = - 2$

The final answer is $- 2$ .

$- 2$

Step 4

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

$(2, - 2)$

Step 5

Find the Maximum/Minimum Value g(x)=-3x^2+12x-14

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