Find the Maximum/Minimum Value f(x)=6x^2+36x+58

$f (x) = 6 x^{2} + 36 x + 58$

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 $b$ .

$x = - \frac{36}{2 (6)}$

Remove parentheses.

$x = - \frac{36}{2 (6)}$

Simplify $- \frac{36}{2 (6)}$ .

Cancel the common factor of $36$ and $2$ .

Factor $2$ out of $36$ .

$x = - \frac{2 \cdot 18}{2 \cdot 6}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 6$ .

$x = - \frac{2 \cdot 18}{2 (6)}$

Cancel the common factor.

$x = - \frac{2 \cdot 18}{2 \cdot 6}$

Rewrite the expression.

$x = - \frac{18}{6}$

Cancel the common factor of $18$ and $6$ .

Factor $6$ out of $18$ .

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

Cancel the common factors.

Factor $6$ out of $6$ .

$x = - \frac{6 \cdot 3}{6 (1)}$

Cancel the common factor.

$x = - \frac{6 \cdot 3}{6 \cdot 1}$

Rewrite the expression.

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

Divide $3$ by $1$ .

$x = - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$x = - 3$

Step 3

Evaluate $f (- 3)$ .

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = 6 {(- 3)}^{2} + 36 (- 3) + 58$

Simplify the result.

Simplify each term.

Raise $- 3$ to the power of $2$ .

$f (- 3) = 6 \cdot 9 + 36 (- 3) + 58$

Multiply $6$ by $9$ .

$f (- 3) = 54 + 36 (- 3) + 58$

Multiply $36$ by $- 3$ .

$f (- 3) = 54 - 108 + 58$

Simplify by adding and subtracting.

Subtract $108$ from $54$ .

$f (- 3) = - 54 + 58$

Add $- 54$ and $58$ .

$f (- 3) = 4$

The final answer is $4$ .

$4$

Step 4

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

$(- 3, 4)$

Step 5

Find the Maximum/Minimum Value f(x)=6x^2+36x+58

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