Precalculus Examples

$\frac{2}{3} n - \frac{1}{90} n^{2}$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{2}{3}$ and $n$ .

$f (n) = \frac{2 n}{3} - \frac{1}{90} n^{2}$

Step 1.2

Combine $n^{2}$ and $\frac{1}{90}$ .

$f (n) = \frac{2 n}{3} - \frac{n^{2}}{90}$

Step 2

The maximum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is negative, the maximum value of the function is $f (- \frac{b}{2 a})$ .

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

Step 3

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

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Step 3.1

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

$n = - \frac{0.\overline{6}}{2 (- 0.0\overline{1})}$

Step 3.2

Remove parentheses.

$n = - \frac{0.\overline{6}}{2 (- 0.0\overline{1})}$

Step 3.3

Simplify $- \frac{0.\overline{6}}{2 (- 0.0\overline{1})}$ .

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Step 3.3.1

Multiply $2$ by $- 0.0\overline{1}$ .

$n = - \frac{0.\overline{6}}{- 0.0\overline{2}}$

Step 3.3.2

Divide $0.\overline{6}$ by $- 0.0\overline{2}$ .

$n = - - 30$

Step 3.3.3

Multiply $- 1$ by $- 30$ .

$n = 30$

Step 4

Evaluate $f (30)$ .

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Step 4.1

Replace the variable $n$ with $30$ in the expression.

$f (30) = \frac{2 (30)}{3} - \frac{{(30)}^{2}}{90}$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $2$ by $30$ .

$f (30) = \frac{60}{3} - \frac{{(30)}^{2}}{90}$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Divide $60$ by $3$ .

$f (30) = 20 - \frac{{(30)}^{2}}{90}$

Step 4.2.2.2

Raise $30$ to the power of $2$ .

$f (30) = 20 - \frac{900}{90}$

Step 4.2.2.3

Divide $900$ by $90$ .

$f (30) = 20 - 1 \cdot 10$

Step 4.2.2.4

Multiply $- 1$ by $10$ .

$f (30) = 20 - 10$

Step 4.2.3

Subtract $10$ from $20$ .

$f (30) = 10$

Step 4.2.4

The final answer is $10$ .

$10$

Step 5

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

$(30, 10)$

Step 6