Finite Math Examples

y = - \frac{1}{8} x^{2} + x

$y = - \frac{1}{8} x^{2} + x$

Step 1

Combine

x^{2}

$x^{2}$ and

\frac{1}{8}

$\frac{1}{8}$ .

f (x) = - \frac{x^{2}}{8} + x

$f (x) = - \frac{x^{2}}{8} + x$

Step 2

The maximum of a quadratic function occurs at

x = - \frac{b}{2 a}

$x = - \frac{b}{2 a}$ . If

a

$a$ is negative, the maximum value of the function is

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

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

f_{max}

$f_{max}$

x = a x^{2} + b x + c

$x = a x^{2} + b x + c$ occurs at

x = - \frac{b}{2 a}

$x = - \frac{b}{2 a}$

Step 3

Find the value of

x = - \frac{b}{2 a}

$x = - \frac{b}{2 a}$ .

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Step 3.1

Substitute in the values of

a

$a$ and

b

$b$ .

x = - \frac{1}{2 (- 0.125)}

$x = - \frac{1}{2 (- 0.125)}$

Step 3.2

Remove parentheses.

x = - \frac{1}{2 (- 0.125)}

$x = - \frac{1}{2 (- 0.125)}$

Step 3.3

Simplify

- \frac{1}{2 (- 0.125)}

$- \frac{1}{2 (- 0.125)}$ .

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Step 3.3.1

Multiply

2

$2$ by

- 0.125

$- 0.125$ .

x = - \frac{1}{- 0.25}

$x = - \frac{1}{- 0.25}$

Step 3.3.2

Divide

1

$1$ by

- 0.25

$- 0.25$ .

x = - - 4

$x = - - 4$

Step 3.3.3

Multiply

- 1

$- 1$ by

- 4

$- 4$ .

x = 4

$x = 4$

x = 4

$x = 4$

x = 4

$x = 4$

Step 4

Evaluate

f (4)

$f (4)$ .

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Step 4.1

Replace the variable

x

$x$ with

4

$4$ in the expression.

f (4) = - \frac{{(4)}^{2}}{8} + 4

$f (4) = - \frac{{(4)}^{2}}{8} + 4$

Step 4.2

Simplify the result.

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Step 4.2.1

Remove parentheses.

f (4) = - \frac{{(4)}^{2}}{8} + 4

$f (4) = - \frac{{(4)}^{2}}{8} + 4$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Raise

4

$4$ to the power of

2

$2$ .

f (4) = - \frac{16}{8} + 4

$f (4) = - \frac{16}{8} + 4$

Step 4.2.2.2

Divide

16

$16$ by

8

$8$ .

f (4) = - 1 \cdot 2 + 4

$f (4) = - 1 \cdot 2 + 4$

Step 4.2.2.3

Multiply

- 1

$- 1$ by

2

$2$ .

f (4) = - 2 + 4

$f (4) = - 2 + 4$

f (4) = - 2 + 4

$f (4) = - 2 + 4$

Step 4.2.3

Add

- 2

$- 2$ and

4

$4$ .

f (4) = 2

$f (4) = 2$

Step 4.2.4

The final answer is

2

$2$ .

2

$2$

2

$2$

2

$2$

Step 5

Use the

x

$x$ and

y

$y$ values to find where the maximum occurs.

(4, 2)

$(4, 2)$

Step 6

⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$