Calculus Examples

$5 x - 3 {(x - 1)}^{2}$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$f (x) = 5 x - 3 ((x - 1) (x - 1))$

Step 1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = 5 x - 3 (x (x - 1) - 1 (x - 1))$

Step 1.2.2

Apply the distributive property.

$f (x) = 5 x - 3 (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Step 1.2.3

Apply the distributive property.

$f (x) = 5 x - 3 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$f (x) = 5 x - 3 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Step 1.3.1.2

Move $- 1$ to the left of $x$ .

$f (x) = 5 x - 3 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

Step 1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$f (x) = 5 x - 3 (x^{2} - x - 1 x - 1 \cdot - 1)$

Step 1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$f (x) = 5 x - 3 (x^{2} - x - x - 1 \cdot - 1)$

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

$f (x) = 5 x - 3 (x^{2} - x - x + 1)$

Step 1.3.2

Subtract $x$ from $- x$ .

$f (x) = 5 x - 3 (x^{2} - 2 x + 1)$

Step 1.4

Apply the distributive property.

$f (x) = 5 x - 3 x^{2} - 3 (- 2 x) - 3 \cdot 1$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 2$ by $- 3$ .

$f (x) = 5 x - 3 x^{2} + 6 x - 3 \cdot 1$

Step 1.5.2

Multiply $- 3$ by $1$ .

$f (x) = 5 x - 3 x^{2} + 6 x - 3$

Step 2

Add $5 x$ and $6 x$ .

$f (x) = - 3 x^{2} + 11 x - 3$

Step 3

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 4

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

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Step 4.1

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

$x = - \frac{11}{2 (- 3)}$

Step 4.2

Remove parentheses.

$x = - \frac{11}{2 (- 3)}$

Step 4.3

Simplify $- \frac{11}{2 (- 3)}$ .

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Step 4.3.1

Multiply $2$ by $- 3$ .

$x = - \frac{11}{- 6}$

Step 4.3.2

Move the negative in front of the fraction.

$x = - - \frac{11}{6}$

Step 4.3.3

Multiply $- - \frac{11}{6}$ .

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Step 4.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{11}{6})$

Step 4.3.3.2

Multiply $\frac{11}{6}$ by $1$ .

$x = \frac{11}{6}$

Step 5

Evaluate $f (\frac{11}{6})$ .

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Step 5.1

Replace the variable $x$ with $\frac{11}{6}$ in the expression.

$f (\frac{11}{6}) = - 3 {(\frac{11}{6})}^{2} + 11 (\frac{11}{6}) - 3$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Apply the product rule to $\frac{11}{6}$ .

$f (\frac{11}{6}) = - 3 \frac{11^{2}}{6^{2}} + 11 (\frac{11}{6}) - 3$

Step 5.2.1.2

Raise $11$ to the power of $2$ .

$f (\frac{11}{6}) = - 3 \frac{121}{6^{2}} + 11 (\frac{11}{6}) - 3$

Step 5.2.1.3

Raise $6$ to the power of $2$ .

$f (\frac{11}{6}) = - 3 (\frac{121}{36}) + 11 (\frac{11}{6}) - 3$

Step 5.2.1.4

Cancel the common factor of $3$ .

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Step 5.2.1.4.1

Factor $3$ out of $- 3$ .

$f (\frac{11}{6}) = 3 (- 1) (\frac{121}{36}) + 11 (\frac{11}{6}) - 3$

Step 5.2.1.4.2

Factor $3$ out of $36$ .

$f (\frac{11}{6}) = 3 \cdot (- 1 \frac{121}{3 \cdot 12}) + 11 (\frac{11}{6}) - 3$

Step 5.2.1.4.3

Cancel the common factor.

$f (\frac{11}{6}) = 3 \cdot (- 1 \frac{121}{3 \cdot 12}) + 11 (\frac{11}{6}) - 3$

Step 5.2.1.4.4

Rewrite the expression.

$f (\frac{11}{6}) = - 1 (\frac{121}{12}) + 11 (\frac{11}{6}) - 3$

Step 5.2.1.5

Rewrite $- 1 (\frac{121}{12})$ as $- (\frac{121}{12})$ .

$f (\frac{11}{6}) = - (\frac{121}{12}) + 11 (\frac{11}{6}) - 3$

Step 5.2.1.6

Multiply $11 (\frac{11}{6})$ .

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Step 5.2.1.6.1

Combine $11$ and $\frac{11}{6}$ .

$f (\frac{11}{6}) = - \frac{121}{12} + \frac{11 \cdot 11}{6} - 3$

Step 5.2.1.6.2

Multiply $11$ by $11$ .

$f (\frac{11}{6}) = - \frac{121}{12} + \frac{121}{6} - 3$

Step 5.2.2

Find the common denominator.

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Step 5.2.2.1

Multiply $\frac{121}{6}$ by $\frac{2}{2}$ .

$f (\frac{11}{6}) = - \frac{121}{12} + \frac{121}{6} \cdot \frac{2}{2} - 3$

Step 5.2.2.2

Multiply $\frac{121}{6}$ by $\frac{2}{2}$ .

$f (\frac{11}{6}) = - \frac{121}{12} + \frac{121 \cdot 2}{6 \cdot 2} - 3$

Step 5.2.2.3

Write $- 3$ as a fraction with denominator $1$ .

$f (\frac{11}{6}) = - \frac{121}{12} + \frac{121 \cdot 2}{6 \cdot 2} + \frac{- 3}{1}$

Step 5.2.2.4

Multiply $\frac{- 3}{1}$ by $\frac{12}{12}$ .

$f (\frac{11}{6}) = - \frac{121}{12} + \frac{121 \cdot 2}{6 \cdot 2} + \frac{- 3}{1} \cdot \frac{12}{12}$

Step 5.2.2.5

Multiply $\frac{- 3}{1}$ by $\frac{12}{12}$ .

$f (\frac{11}{6}) = - \frac{121}{12} + \frac{121 \cdot 2}{6 \cdot 2} + \frac{- 3 \cdot 12}{12}$

Step 5.2.2.6

Reorder the factors of $6 \cdot 2$ .

$f (\frac{11}{6}) = - \frac{121}{12} + \frac{121 \cdot 2}{2 \cdot 6} + \frac{- 3 \cdot 12}{12}$

Step 5.2.2.7

Multiply $2$ by $6$ .

$f (\frac{11}{6}) = - \frac{121}{12} + \frac{121 \cdot 2}{12} + \frac{- 3 \cdot 12}{12}$

Step 5.2.3

Combine the numerators over the common denominator.

$f (\frac{11}{6}) = \frac{- 121 + 121 \cdot 2 - 3 \cdot 12}{12}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Multiply $121$ by $2$ .

$f (\frac{11}{6}) = \frac{- 121 + 242 - 3 \cdot 12}{12}$

Step 5.2.4.2

Multiply $- 3$ by $12$ .

$f (\frac{11}{6}) = \frac{- 121 + 242 - 36}{12}$

Step 5.2.5

Simplify by adding and subtracting.

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Step 5.2.5.1

Add $- 121$ and $242$ .

$f (\frac{11}{6}) = \frac{121 - 36}{12}$

Step 5.2.5.2

Subtract $36$ from $121$ .

$f (\frac{11}{6}) = \frac{85}{12}$

Step 5.2.6

The final answer is $\frac{85}{12}$ .

$\frac{85}{12}$

Step 6

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

$(\frac{11}{6}, \frac{85}{12})$

Step 7