Finite Math Examples

$f (x) = - 7 x^{2} + 6 x + 3$

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

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Step 2.1

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

$x = - \frac{6}{2 (- 7)}$

Step 2.2

Remove parentheses.

$x = - \frac{6}{2 (- 7)}$

Step 2.3

Simplify $- \frac{6}{2 (- 7)}$ .

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Step 2.3.1

Cancel the common factor of $6$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $6$ .

$x = - \frac{2 \cdot 3}{2 \cdot - 7}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2 \cdot - 7$ .

$x = - \frac{2 \cdot 3}{2 (- 7)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot 3}{2 \cdot - 7}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{3}{- 7}$

Step 2.3.2

Move the negative in front of the fraction.

$x = - - \frac{3}{7}$

Step 2.3.3

Multiply $- - \frac{3}{7}$ .

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{3}{7})$

Step 2.3.3.2

Multiply $\frac{3}{7}$ by $1$ .

$x = \frac{3}{7}$

Step 3

Evaluate $f (\frac{3}{7})$ .

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Step 3.1

Replace the variable $x$ with $\frac{3}{7}$ in the expression.

$f (\frac{3}{7}) = - 7 {(\frac{3}{7})}^{2} + 6 (\frac{3}{7}) + 3$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the product rule to $\frac{3}{7}$ .

$f (\frac{3}{7}) = - 7 \frac{3^{2}}{7^{2}} + 6 (\frac{3}{7}) + 3$

Step 3.2.1.2

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

$f (\frac{3}{7}) = - 7 \frac{9}{7^{2}} + 6 (\frac{3}{7}) + 3$

Step 3.2.1.3

Raise $7$ to the power of $2$ .

$f (\frac{3}{7}) = - 7 (\frac{9}{49}) + 6 (\frac{3}{7}) + 3$

Step 3.2.1.4

Cancel the common factor of $7$ .

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Step 3.2.1.4.1

Factor $7$ out of $- 7$ .

$f (\frac{3}{7}) = 7 (- 1) (\frac{9}{49}) + 6 (\frac{3}{7}) + 3$

Step 3.2.1.4.2

Factor $7$ out of $49$ .

$f (\frac{3}{7}) = 7 \cdot (- 1 \frac{9}{7 \cdot 7}) + 6 (\frac{3}{7}) + 3$

Step 3.2.1.4.3

Cancel the common factor.

$f (\frac{3}{7}) = 7 \cdot (- 1 \frac{9}{7 \cdot 7}) + 6 (\frac{3}{7}) + 3$

Step 3.2.1.4.4

Rewrite the expression.

$f (\frac{3}{7}) = - 1 (\frac{9}{7}) + 6 (\frac{3}{7}) + 3$

Step 3.2.1.5

Rewrite $- 1 (\frac{9}{7})$ as $- (\frac{9}{7})$ .

$f (\frac{3}{7}) = - (\frac{9}{7}) + 6 (\frac{3}{7}) + 3$

Step 3.2.1.6

Multiply $6 (\frac{3}{7})$ .

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Step 3.2.1.6.1

Combine $6$ and $\frac{3}{7}$ .

$f (\frac{3}{7}) = - \frac{9}{7} + \frac{6 \cdot 3}{7} + 3$

Step 3.2.1.6.2

Multiply $6$ by $3$ .

$f (\frac{3}{7}) = - \frac{9}{7} + \frac{18}{7} + 3$

Step 3.2.2

Combine fractions.

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Step 3.2.2.1

Combine the numerators over the common denominator.

$f (\frac{3}{7}) = 3 + \frac{- 9 + 18}{7}$

Step 3.2.2.2

Add $- 9$ and $18$ .

$f (\frac{3}{7}) = 3 + \frac{9}{7}$

Step 3.2.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$f (\frac{3}{7}) = 3 \cdot \frac{7}{7} + \frac{9}{7}$

Step 3.2.4

Combine $3$ and $\frac{7}{7}$ .

$f (\frac{3}{7}) = \frac{3 \cdot 7}{7} + \frac{9}{7}$

Step 3.2.5

Combine the numerators over the common denominator.

$f (\frac{3}{7}) = \frac{3 \cdot 7 + 9}{7}$

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $3$ by $7$ .

$f (\frac{3}{7}) = \frac{21 + 9}{7}$

Step 3.2.6.2

Add $21$ and $9$ .

$f (\frac{3}{7}) = \frac{30}{7}$

Step 3.2.7

The final answer is $\frac{30}{7}$ .

$\frac{30}{7}$

Step 4

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

$(\frac{3}{7}, \frac{30}{7})$

Step 5