Linear Algebra Examples

$f (x) = - 2 x^{2} + 9 x + 1$

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 $b$ .

$x = - \frac{9}{2 (- 2)}$

Step 2.2

Remove parentheses.

$x = - \frac{9}{2 (- 2)}$

Step 2.3

Simplify $- \frac{9}{2 (- 2)}$ .

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Step 2.3.1

Multiply $2$ by $- 2$ .

$x = - \frac{9}{- 4}$

Step 2.3.2

Move the negative in front of the fraction.

$x = - - \frac{9}{4}$

Step 2.3.3

Multiply $- - \frac{9}{4}$ .

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{9}{4})$

Step 2.3.3.2

Multiply $\frac{9}{4}$ by $1$ .

$x = \frac{9}{4}$

Step 3

Evaluate $f (\frac{9}{4})$ .

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Step 3.1

Replace the variable $x$ with $\frac{9}{4}$ in the expression.

$f (\frac{9}{4}) = - 2 {(\frac{9}{4})}^{2} + 9 (\frac{9}{4}) + 1$

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 $\frac{9}{4}$ .

$f (\frac{9}{4}) = - 2 \frac{9^{2}}{4^{2}} + 9 (\frac{9}{4}) + 1$

Step 3.2.1.2

Raise $9$ to the power of $2$ .

$f (\frac{9}{4}) = - 2 \frac{81}{4^{2}} + 9 (\frac{9}{4}) + 1$

Step 3.2.1.3

Raise $4$ to the power of $2$ .

$f (\frac{9}{4}) = - 2 (\frac{81}{16}) + 9 (\frac{9}{4}) + 1$

Step 3.2.1.4

Cancel the common factor of $2$ .

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Step 3.2.1.4.1

Factor $2$ out of $- 2$ .

$f (\frac{9}{4}) = 2 (- 1) (\frac{81}{16}) + 9 (\frac{9}{4}) + 1$

Step 3.2.1.4.2

Factor $2$ out of $16$ .

$f (\frac{9}{4}) = 2 \cdot (- 1 \frac{81}{2 \cdot 8}) + 9 (\frac{9}{4}) + 1$

Step 3.2.1.4.3

Cancel the common factor.

$f (\frac{9}{4}) = 2 \cdot (- 1 \frac{81}{2 \cdot 8}) + 9 (\frac{9}{4}) + 1$

Step 3.2.1.4.4

Rewrite the expression.

$f (\frac{9}{4}) = - 1 (\frac{81}{8}) + 9 (\frac{9}{4}) + 1$

Step 3.2.1.5

Rewrite $- 1 (\frac{81}{8})$ as $- (\frac{81}{8})$ .

$f (\frac{9}{4}) = - (\frac{81}{8}) + 9 (\frac{9}{4}) + 1$

Step 3.2.1.6

Multiply $9 (\frac{9}{4})$ .

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Step 3.2.1.6.1

Combine $9$ and $\frac{9}{4}$ .

$f (\frac{9}{4}) = - \frac{81}{8} + \frac{9 \cdot 9}{4} + 1$

Step 3.2.1.6.2

Multiply $9$ by $9$ .

$f (\frac{9}{4}) = - \frac{81}{8} + \frac{81}{4} + 1$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Multiply $\frac{81}{4}$ by $\frac{2}{2}$ .

$f (\frac{9}{4}) = - \frac{81}{8} + \frac{81}{4} \cdot \frac{2}{2} + 1$

Step 3.2.2.2

Multiply $\frac{81}{4}$ by $\frac{2}{2}$ .

$f (\frac{9}{4}) = - \frac{81}{8} + \frac{81 \cdot 2}{4 \cdot 2} + 1$

Step 3.2.2.3

Write $1$ as a fraction with denominator $1$ .

$f (\frac{9}{4}) = - \frac{81}{8} + \frac{81 \cdot 2}{4 \cdot 2} + \frac{1}{1}$

Step 3.2.2.4

Multiply $\frac{1}{1}$ by $\frac{8}{8}$ .

$f (\frac{9}{4}) = - \frac{81}{8} + \frac{81 \cdot 2}{4 \cdot 2} + \frac{1}{1} \cdot \frac{8}{8}$

Step 3.2.2.5

Multiply $\frac{1}{1}$ by $\frac{8}{8}$ .

$f (\frac{9}{4}) = - \frac{81}{8} + \frac{81 \cdot 2}{4 \cdot 2} + \frac{8}{8}$

Step 3.2.2.6

Reorder the factors of $4 \cdot 2$ .

$f (\frac{9}{4}) = - \frac{81}{8} + \frac{81 \cdot 2}{2 \cdot 4} + \frac{8}{8}$

Step 3.2.2.7

Multiply $2$ by $4$ .

$f (\frac{9}{4}) = - \frac{81}{8} + \frac{81 \cdot 2}{8} + \frac{8}{8}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (\frac{9}{4}) = \frac{- 81 + 81 \cdot 2 + 8}{8}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Multiply $81$ by $2$ .

$f (\frac{9}{4}) = \frac{- 81 + 162 + 8}{8}$

Step 3.2.4.2

Add $- 81$ and $162$ .

$f (\frac{9}{4}) = \frac{81 + 8}{8}$

Step 3.2.4.3

Add $81$ and $8$ .

$f (\frac{9}{4}) = \frac{89}{8}$

Step 3.2.5

The final answer is $\frac{89}{8}$ .

$\frac{89}{8}$

Step 4

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

$(\frac{9}{4}, \frac{89}{8})$

Step 5