Find the Maximum/Minimum Value f(x)=-6x^2+312x-3762

$f (x) = - 6 x^{2} + 312 x - 3762$

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 $b$ .

$x = - \frac{312}{2 (- 6)}$

Remove parentheses.

$x = - \frac{312}{2 (- 6)}$

Simplify $- \frac{312}{2 (- 6)}$ .

Cancel the common factor of $312$ and $2$ .

Factor $2$ out of $312$ .

$x = - \frac{2 \cdot 156}{2 \cdot - 6}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 156}{2 (- 6)}$

Cancel the common factor.

$x = - \frac{2 \cdot 156}{2 \cdot - 6}$

Rewrite the expression.

$x = - \frac{156}{- 6}$

Cancel the common factor of $156$ and $- 6$ .

Factor $6$ out of $156$ .

$x = - \frac{6 (26)}{- 6}$

Move the negative one from the denominator of $\frac{26}{- 1}$ .

$x = - (- 1 \cdot 26)$

Multiply $- (- 1 \cdot 26)$ .

Multiply $- 1$ by $26$ .

$x = - - 26$

Multiply $- 1$ by $- 26$ .

$x = 26$

Step 3

Evaluate $f (26)$ .

Replace the variable $x$ with $26$ in the expression.

$f (26) = - 6 {(26)}^{2} + 312 (26) - 3762$

Simplify the result.

Simplify each term.

Raise $26$ to the power of $2$ .

$f (26) = - 6 \cdot 676 + 312 (26) - 3762$

Multiply $- 6$ by $676$ .

$f (26) = - 4056 + 312 (26) - 3762$

Multiply $312$ by $26$ .

$f (26) = - 4056 + 8112 - 3762$

Simplify by adding and subtracting.

Add $- 4056$ and $8112$ .

$f (26) = 4056 - 3762$

Subtract $3762$ from $4056$ .

$f (26) = 294$

The final answer is $294$ .

$294$

Step 4

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

$(26, 294)$

Step 5

Find the Maximum/Minimum Value f(x)=-6x^2+312x-3762

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