Precalculus Examples

$h (t) = 36 t - 6 t^{2}$

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 $f (- \frac{b}{2 a})$ .

$f_{max}$ $t = a t^{2} + b t + c$ occurs at $t = - \frac{b}{2 a}$

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

Remove parentheses.

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Factor $2$ out of $36$ .

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

Step 2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2

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

Tap for more steps...

Step 2.3.2.1

Factor $6$ out of $18$ .

$t = - \frac{6 (3)}{- 6}$

Step 2.3.2.2

Move the negative one from the denominator of $\frac{3}{- 1}$ .

$t = - (- 1 \cdot 3)$

Step 2.3.3

Multiply $- (- 1 \cdot 3)$ .

Tap for more steps...

Step 2.3.3.1

Multiply $- 1$ by $3$ .

$t = - - 3$

Step 2.3.3.2

Multiply $- 1$ by $- 3$ .

$t = 3$

Step 3

Evaluate $f (3)$ .

Tap for more steps...

Step 3.1

Replace the variable $t$ with $3$ in the expression.

$h (3) = 36 (3) - 6 {(3)}^{2}$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Multiply $36$ by $3$ .

$h (3) = 108 - 6 {(3)}^{2}$

Step 3.2.1.2

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

$h (3) = 108 - 6 \cdot 9$

Step 3.2.1.3

Multiply $- 6$ by $9$ .

$h (3) = 108 - 54$

Step 3.2.2

Subtract $54$ from $108$ .

$h (3) = 54$

Step 3.2.3

The final answer is $54$ .

$54$

Step 4

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

$(3, 54)$

Step 5