Calculus Examples

$t (x) = 101 - \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $101 - \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))$ with respect to $x$ is $\frac{d}{d x} [101] + \frac{d}{d x} [- \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))]$ .

$\frac{d}{d x} [101] + \frac{d}{d x} [- \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))]$

Step 1.1.2

Since $101$ is constant with respect to $x$ , the derivative of $101$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))]$

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))]$ .

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Step 1.2.1

Combine $x^{2}$ and $\frac{1}{6}$ .

$0 + \frac{d}{d x} [- \frac{x^{2}}{6} \cdot (1 - \frac{x}{6})]$

Step 1.2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{6} (1 - \frac{x}{6})$ with respect to $x$ is $- \frac{d}{d x} [\frac{x^{2}}{6} (1 - \frac{x}{6})]$ .

$0 - \frac{d}{d x} [\frac{x^{2}}{6} (1 - \frac{x}{6})]$

Step 1.2.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \frac{x^{2}}{6}$ and $g (x) = 1 - \frac{x}{6}$ .

$0 - (\frac{x^{2}}{6} \frac{d}{d x} [1 - \frac{x}{6}] + (1 - \frac{x}{6}) \frac{d}{d x} [\frac{x^{2}}{6}])$

Step 1.2.4

By the Sum Rule, the derivative of $1 - \frac{x}{6}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{6}]$ .

$0 - (\frac{x^{2}}{6} (\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{6}]) + (1 - \frac{x}{6}) \frac{d}{d x} [\frac{x^{2}}{6}])$

Step 1.2.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 - (\frac{x^{2}}{6} (0 + \frac{d}{d x} [- \frac{x}{6}]) + (1 - \frac{x}{6}) \frac{d}{d x} [\frac{x^{2}}{6}])$

Step 1.2.6

Since $- \frac{1}{6}$ is constant with respect to $x$ , the derivative of $- \frac{x}{6}$ with respect to $x$ is $- \frac{1}{6} \frac{d}{d x} [x]$ .

$0 - (\frac{x^{2}}{6} (0 - \frac{1}{6} \frac{d}{d x} [x]) + (1 - \frac{x}{6}) \frac{d}{d x} [\frac{x^{2}}{6}])$

Step 1.2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$0 - (\frac{x^{2}}{6} (0 - \frac{1}{6} \cdot 1) + (1 - \frac{x}{6}) \frac{d}{d x} [\frac{x^{2}}{6}])$

Step 1.2.8

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{6}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x^{2}]$ .

$0 - (\frac{x^{2}}{6} (0 - \frac{1}{6} \cdot 1) + (1 - \frac{x}{6}) (\frac{1}{6} \frac{d}{d x} [x^{2}]))$

Step 1.2.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - (\frac{x^{2}}{6} (0 - \frac{1}{6} \cdot 1) + (1 - \frac{x}{6}) (\frac{1}{6} (2 x)))$

Step 1.2.10

Multiply $- 1$ by $1$ .

$0 - (\frac{x^{2}}{6} (0 - \frac{1}{6}) + (1 - \frac{x}{6}) (\frac{1}{6} (2 x)))$

Step 1.2.11

Subtract $\frac{1}{6}$ from $0$ .

$0 - (\frac{x^{2}}{6} (- \frac{1}{6}) + (1 - \frac{x}{6}) (\frac{1}{6} (2 x)))$

Step 1.2.12

Multiply $\frac{x^{2}}{6}$ by $\frac{1}{6}$ .

$0 - (- \frac{x^{2}}{6 \cdot 6} + (1 - \frac{x}{6}) (\frac{1}{6} (2 x)))$

Step 1.2.13

Multiply $6$ by $6$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) (\frac{1}{6} (2 x)))$

Step 1.2.14

Combine $2$ and $\frac{1}{6}$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) (\frac{2}{6} x))$

Step 1.2.15

Combine $\frac{2}{6}$ and $x$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{2 x}{6})$

Step 1.2.16

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

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Step 1.2.16.1

Factor $2$ out of $2 x$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{2 (x)}{6})$

Step 1.2.16.2

Cancel the common factors.

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Step 1.2.16.2.1

Factor $2$ out of $6$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{2 x}{2 \cdot 3})$

Step 1.2.16.2.2

Cancel the common factor.

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{2 x}{2 \cdot 3})$

Step 1.2.16.2.3

Rewrite the expression.

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{x}{3})$

Step 1.2.17

To write $(1 - \frac{x}{6}) \frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{36}{36}$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{x}{3} \cdot \frac{36}{36})$

Step 1.2.18

Combine $(1 - \frac{x}{6}) \frac{x}{3}$ and $\frac{36}{36}$ .

$0 - (- \frac{x^{2}}{36} + \frac{(1 - \frac{x}{6}) \frac{x}{3} \cdot 36}{36})$

Step 1.2.19

Combine the numerators over the common denominator.

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{x}{3} \cdot 36}{36}$

Step 1.2.20

Combine $36$ and $\frac{x}{3}$ .

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{36 x}{3}}{36}$

Step 1.2.21

Cancel the common factor of $36$ and $3$ .

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Step 1.2.21.1

Factor $3$ out of $36 x$ .

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{3 (12 x)}{3}}{36}$

Step 1.2.21.2

Cancel the common factors.

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Step 1.2.21.2.1

Factor $3$ out of $3$ .

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{3 (12 x)}{3 (1)}}{36}$

Step 1.2.21.2.2

Cancel the common factor.

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{3 (12 x)}{3 \cdot 1}}{36}$

Step 1.2.21.2.3

Rewrite the expression.

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{12 x}{1}}{36}$

Step 1.2.21.2.4

Divide $12 x$ by $1$ .

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) (12 x)}{36}$

Step 1.2.22

Move $12$ to the left of $1 - \frac{x}{6}$ .

$0 - \frac{- x^{2} + 12 (1 - \frac{x}{6}) x}{36}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$0 - \frac{- x^{2} + (12 \cdot 1 + 12 (- \frac{x}{6})) x}{36}$

Step 1.3.2

Apply the distributive property.

$0 - \frac{- x^{2} + 12 \cdot 1 x + 12 (- \frac{x}{6}) x}{36}$

Step 1.3.3

Combine terms.

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Step 1.3.3.1

Multiply $12$ by $1$ .

$0 - \frac{- x^{2} + 12 x + 12 (- \frac{x}{6}) x}{36}$

Step 1.3.3.2

Multiply $- 1$ by $12$ .

$0 - \frac{- x^{2} + 12 x - 12 \frac{x}{6} x}{36}$

Step 1.3.3.3

Combine $- 12$ and $\frac{x}{6}$ .

$0 - \frac{- x^{2} + 12 x + \frac{- 12 x}{6} x}{36}$

Step 1.3.3.4

Cancel the common factor of $- 12$ and $6$ .

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Step 1.3.3.4.1

Factor $6$ out of $- 12 x$ .

$0 - \frac{- x^{2} + 12 x + \frac{6 (- 2 x)}{6} x}{36}$

Step 1.3.3.4.2

Cancel the common factors.

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Step 1.3.3.4.2.1

Factor $6$ out of $6$ .

$0 - \frac{- x^{2} + 12 x + \frac{6 (- 2 x)}{6 (1)} x}{36}$

Step 1.3.3.4.2.2

Cancel the common factor.

$0 - \frac{- x^{2} + 12 x + \frac{6 (- 2 x)}{6 \cdot 1} x}{36}$

Step 1.3.3.4.2.3

Rewrite the expression.

$0 - \frac{- x^{2} + 12 x + \frac{- 2 x}{1} x}{36}$

Step 1.3.3.4.2.4

Divide $- 2 x$ by $1$ .

$0 - \frac{- x^{2} + 12 x - 2 x \cdot x}{36}$

Step 1.3.3.5

Raise $x$ to the power of $1$ .

$0 - \frac{- x^{2} + 12 x - 2 (x^{1} x)}{36}$

Step 1.3.3.6

Raise $x$ to the power of $1$ .

$0 - \frac{- x^{2} + 12 x - 2 (x^{1} x^{1})}{36}$

Step 1.3.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$0 - \frac{- x^{2} + 12 x - 2 x^{1 + 1}}{36}$

Step 1.3.3.8

Add $1$ and $1$ .

$0 - \frac{- x^{2} + 12 x - 2 x^{2}}{36}$

Step 1.3.3.9

Subtract $2 x^{2}$ from $- x^{2}$ .

$0 - \frac{- 3 x^{2} + 12 x}{36}$

Step 1.3.3.10

Subtract $\frac{- 3 x^{2} + 12 x}{36}$ from $0$ .

$- \frac{- 3 x^{2} + 12 x}{36}$

Step 1.3.4

Factor $3 x$ out of $- 3 x^{2} + 12 x$ .

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Step 1.3.4.1

Factor $3 x$ out of $- 3 x^{2}$ .

$- \frac{3 x (- x) + 12 x}{36}$

Step 1.3.4.2

Factor $3 x$ out of $12 x$ .

$- \frac{3 x (- x) + 3 x (4)}{36}$

Step 1.3.4.3

Factor $3 x$ out of $3 x (- x) + 3 x (4)$ .

$- \frac{3 x (- x + 4)}{36}$

Step 1.3.5

Cancel the common factor of $3$ and $36$ .

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Step 1.3.5.1

Factor $3$ out of $3 x (- x + 4)$ .

$- \frac{3 (x (- x + 4))}{36}$

Step 1.3.5.2

Cancel the common factors.

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Step 1.3.5.2.1

Factor $3$ out of $36$ .

$- \frac{3 (x (- x + 4))}{3 \cdot 12}$

Step 1.3.5.2.2

Cancel the common factor.

$- \frac{3 (x (- x + 4))}{3 \cdot 12}$

Step 1.3.5.2.3

Rewrite the expression.

$- \frac{x (- x + 4)}{12}$

Step 1.3.6

Factor $- 1$ out of $- x$ .

$- \frac{x (- (x) + 4)}{12}$

Step 1.3.7

Rewrite $4$ as $- 1 (- 4)$ .

$- \frac{x (- (x) - 1 (- 4))}{12}$

Step 1.3.8

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$- \frac{x (- (x - 4))}{12}$

Step 1.3.9

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$- \frac{x (- 1 (x - 4))}{12}$

Step 1.3.10

Move the negative in front of the fraction.

$- - \frac{x (x - 4)}{12}$

Step 1.3.11

Multiply $- 1$ by $- 1$ .

$1 \frac{x (x - 4)}{12}$

Step 1.3.12

Multiply $\frac{x (x - 4)}{12}$ by $1$ .

$\frac{x (x - 4)}{12}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{1}{12}$ is constant with respect to $x$ , the derivative of $\frac{x (x - 4)}{12}$ with respect to $x$ is $\frac{1}{12} \frac{d}{d x} [x (x - 4)]$ .

$t'' (x) = \frac{1}{12} \cdot \frac{d}{d x} (x (x - 4))$

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x - 4$ .

$t'' (x) = \frac{1}{12} \cdot (x \frac{d}{d x} (x - 4) + (x - 4) \frac{d}{d x} (x))$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$t'' (x) = \frac{1}{12} \cdot (x (\frac{d}{d x} (x) + \frac{d}{d x} (- 4)) + (x - 4) \frac{d}{d x} (x))$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$t'' (x) = \frac{1}{12} \cdot (x (1 + \frac{d}{d x} (- 4)) + (x - 4) \frac{d}{d x} (x))$

Step 2.3.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$t'' (x) = \frac{1}{12} \cdot (x (1 + 0) + (x - 4) \frac{d}{d x} (x))$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

Add $1$ and $0$ .

$t'' (x) = \frac{1}{12} \cdot (x \cdot 1 + (x - 4) \frac{d}{d x} (x))$

Step 2.3.4.2

Multiply $x$ by $1$ .

$t'' (x) = \frac{1}{12} \cdot (x + (x - 4) \frac{d}{d x} (x))$

Step 2.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$t'' (x) = \frac{1}{12} \cdot (x + (x - 4) \cdot 1)$

Step 2.3.6

Simplify by adding terms.

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Step 2.3.6.1

Multiply $x - 4$ by $1$ .

$t'' (x) = \frac{1}{12} \cdot (x + x - 4)$

Step 2.3.6.2

Add $x$ and $x$ .

$t'' (x) = \frac{1}{12} \cdot (2 x - 4)$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$t'' (x) = \frac{1}{12} \cdot (2 x) + \frac{1}{12} \cdot - 4$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Combine $2$ and $\frac{1}{12}$ .

$t'' (x) = \frac{2}{12} x + \frac{1}{12} \cdot - 4$

Step 2.4.2.2

Combine $\frac{2}{12}$ and $x$ .

$t'' (x) = \frac{2 x}{12} + \frac{1}{12} \cdot - 4$

Step 2.4.2.3

Cancel the common factor of $2$ and $12$ .

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Step 2.4.2.3.1

Factor $2$ out of $2 x$ .

$t'' (x) = \frac{2 (x)}{12} + \frac{1}{12} \cdot - 4$

Step 2.4.2.3.2

Cancel the common factors.

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Step 2.4.2.3.2.1

Factor $2$ out of $12$ .

$t'' (x) = \frac{2 x}{2 \cdot 6} + \frac{1}{12} \cdot - 4$

Step 2.4.2.3.2.2

Cancel the common factor.

$t'' (x) = \frac{2 x}{2 \cdot 6} + \frac{1}{12} \cdot - 4$

Step 2.4.2.3.2.3

Rewrite the expression.

$t'' (x) = \frac{x}{6} + \frac{1}{12} \cdot - 4$

Step 2.4.2.4

Combine $\frac{1}{12}$ and $- 4$ .

$t'' (x) = \frac{x}{6} + \frac{- 4}{12}$

Step 2.4.2.5

Cancel the common factor of $- 4$ and $12$ .

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Step 2.4.2.5.1

Factor $4$ out of $- 4$ .

$t'' (x) = \frac{x}{6} + \frac{4 (- 1)}{12}$

Step 2.4.2.5.2

Cancel the common factors.

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Step 2.4.2.5.2.1

Factor $4$ out of $12$ .

$t'' (x) = \frac{x}{6} + \frac{4 \cdot - 1}{4 \cdot 3}$

Step 2.4.2.5.2.2

Cancel the common factor.

$t'' (x) = \frac{x}{6} + \frac{4 \cdot - 1}{4 \cdot 3}$

Step 2.4.2.5.2.3

Rewrite the expression.

$t'' (x) = \frac{x}{6} + \frac{- 1}{3}$

Step 2.4.2.6

Move the negative in front of the fraction.

$t'' (x) = \frac{x}{6} - \frac{1}{3}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{x (x - 4)}{12} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Differentiate.

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Step 4.1.1.1

By the Sum Rule, the derivative of $101 - \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))$ with respect to $x$ is $\frac{d}{d x} [101] + \frac{d}{d x} [- \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))]$ .

$\frac{d}{d x} [101] + \frac{d}{d x} [- \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))]$

Step 4.1.1.2

Since $101$ is constant with respect to $x$ , the derivative of $101$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))]$

Step 4.1.2

Evaluate $\frac{d}{d x} [- \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))]$ .

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Step 4.1.2.1

Combine $x^{2}$ and $\frac{1}{6}$ .

$0 + \frac{d}{d x} [- \frac{x^{2}}{6} \cdot (1 - \frac{x}{6})]$

Step 4.1.2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{6} (1 - \frac{x}{6})$ with respect to $x$ is $- \frac{d}{d x} [\frac{x^{2}}{6} (1 - \frac{x}{6})]$ .

$0 - \frac{d}{d x} [\frac{x^{2}}{6} (1 - \frac{x}{6})]$

Step 4.1.2.3

$0 - (\frac{x^{2}}{6} \frac{d}{d x} [1 - \frac{x}{6}] + (1 - \frac{x}{6}) \frac{d}{d x} [\frac{x^{2}}{6}])$

Step 4.1.2.4

By the Sum Rule, the derivative of $1 - \frac{x}{6}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{6}]$ .

$0 - (\frac{x^{2}}{6} (\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{6}]) + (1 - \frac{x}{6}) \frac{d}{d x} [\frac{x^{2}}{6}])$

Step 4.1.2.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 - (\frac{x^{2}}{6} (0 + \frac{d}{d x} [- \frac{x}{6}]) + (1 - \frac{x}{6}) \frac{d}{d x} [\frac{x^{2}}{6}])$

Step 4.1.2.6

Since $- \frac{1}{6}$ is constant with respect to $x$ , the derivative of $- \frac{x}{6}$ with respect to $x$ is $- \frac{1}{6} \frac{d}{d x} [x]$ .

$0 - (\frac{x^{2}}{6} (0 - \frac{1}{6} \frac{d}{d x} [x]) + (1 - \frac{x}{6}) \frac{d}{d x} [\frac{x^{2}}{6}])$

Step 4.1.2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$0 - (\frac{x^{2}}{6} (0 - \frac{1}{6} \cdot 1) + (1 - \frac{x}{6}) \frac{d}{d x} [\frac{x^{2}}{6}])$

Step 4.1.2.8

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{6}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x^{2}]$ .

$0 - (\frac{x^{2}}{6} (0 - \frac{1}{6} \cdot 1) + (1 - \frac{x}{6}) (\frac{1}{6} \frac{d}{d x} [x^{2}]))$

Step 4.1.2.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - (\frac{x^{2}}{6} (0 - \frac{1}{6} \cdot 1) + (1 - \frac{x}{6}) (\frac{1}{6} (2 x)))$

Step 4.1.2.10

Multiply $- 1$ by $1$ .

$0 - (\frac{x^{2}}{6} (0 - \frac{1}{6}) + (1 - \frac{x}{6}) (\frac{1}{6} (2 x)))$

Step 4.1.2.11

Subtract $\frac{1}{6}$ from $0$ .

$0 - (\frac{x^{2}}{6} (- \frac{1}{6}) + (1 - \frac{x}{6}) (\frac{1}{6} (2 x)))$

Step 4.1.2.12

Multiply $\frac{x^{2}}{6}$ by $\frac{1}{6}$ .

$0 - (- \frac{x^{2}}{6 \cdot 6} + (1 - \frac{x}{6}) (\frac{1}{6} (2 x)))$

Step 4.1.2.13

Multiply $6$ by $6$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) (\frac{1}{6} (2 x)))$

Step 4.1.2.14

Combine $2$ and $\frac{1}{6}$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) (\frac{2}{6} x))$

Step 4.1.2.15

Combine $\frac{2}{6}$ and $x$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{2 x}{6})$

Step 4.1.2.16

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

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Step 4.1.2.16.1

Factor $2$ out of $2 x$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{2 (x)}{6})$

Step 4.1.2.16.2

Cancel the common factors.

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Step 4.1.2.16.2.1

Factor $2$ out of $6$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{2 x}{2 \cdot 3})$

Step 4.1.2.16.2.2

Cancel the common factor.

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{2 x}{2 \cdot 3})$

Step 4.1.2.16.2.3

Rewrite the expression.

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{x}{3})$

Step 4.1.2.17

To write $(1 - \frac{x}{6}) \frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{36}{36}$ .

$0 - (- \frac{x^{2}}{36} + (1 - \frac{x}{6}) \frac{x}{3} \cdot \frac{36}{36})$

Step 4.1.2.18

Combine $(1 - \frac{x}{6}) \frac{x}{3}$ and $\frac{36}{36}$ .

$0 - (- \frac{x^{2}}{36} + \frac{(1 - \frac{x}{6}) \frac{x}{3} \cdot 36}{36})$

Step 4.1.2.19

Combine the numerators over the common denominator.

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{x}{3} \cdot 36}{36}$

Step 4.1.2.20

Combine $36$ and $\frac{x}{3}$ .

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{36 x}{3}}{36}$

Step 4.1.2.21

Cancel the common factor of $36$ and $3$ .

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Step 4.1.2.21.1

Factor $3$ out of $36 x$ .

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{3 (12 x)}{3}}{36}$

Step 4.1.2.21.2

Cancel the common factors.

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Step 4.1.2.21.2.1

Factor $3$ out of $3$ .

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{3 (12 x)}{3 (1)}}{36}$

Step 4.1.2.21.2.2

Cancel the common factor.

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{3 (12 x)}{3 \cdot 1}}{36}$

Step 4.1.2.21.2.3

Rewrite the expression.

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) \frac{12 x}{1}}{36}$

Step 4.1.2.21.2.4

Divide $12 x$ by $1$ .

$0 - \frac{- x^{2} + (1 - \frac{x}{6}) (12 x)}{36}$

Step 4.1.2.22

Move $12$ to the left of $1 - \frac{x}{6}$ .

$0 - \frac{- x^{2} + 12 (1 - \frac{x}{6}) x}{36}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Apply the distributive property.

$0 - \frac{- x^{2} + (12 \cdot 1 + 12 (- \frac{x}{6})) x}{36}$

Step 4.1.3.2

Apply the distributive property.

$0 - \frac{- x^{2} + 12 \cdot 1 x + 12 (- \frac{x}{6}) x}{36}$

Step 4.1.3.3

Combine terms.

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Step 4.1.3.3.1

Multiply $12$ by $1$ .

$0 - \frac{- x^{2} + 12 x + 12 (- \frac{x}{6}) x}{36}$

Step 4.1.3.3.2

Multiply $- 1$ by $12$ .

$0 - \frac{- x^{2} + 12 x - 12 \frac{x}{6} x}{36}$

Step 4.1.3.3.3

Combine $- 12$ and $\frac{x}{6}$ .

$0 - \frac{- x^{2} + 12 x + \frac{- 12 x}{6} x}{36}$

Step 4.1.3.3.4

Cancel the common factor of $- 12$ and $6$ .

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Step 4.1.3.3.4.1

Factor $6$ out of $- 12 x$ .

$0 - \frac{- x^{2} + 12 x + \frac{6 (- 2 x)}{6} x}{36}$

Step 4.1.3.3.4.2

Cancel the common factors.

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Step 4.1.3.3.4.2.1

Factor $6$ out of $6$ .

$0 - \frac{- x^{2} + 12 x + \frac{6 (- 2 x)}{6 (1)} x}{36}$

Step 4.1.3.3.4.2.2

Cancel the common factor.

$0 - \frac{- x^{2} + 12 x + \frac{6 (- 2 x)}{6 \cdot 1} x}{36}$

Step 4.1.3.3.4.2.3

Rewrite the expression.

$0 - \frac{- x^{2} + 12 x + \frac{- 2 x}{1} x}{36}$

Step 4.1.3.3.4.2.4

Divide $- 2 x$ by $1$ .

$0 - \frac{- x^{2} + 12 x - 2 x \cdot x}{36}$

Step 4.1.3.3.5

Raise $x$ to the power of $1$ .

$0 - \frac{- x^{2} + 12 x - 2 (x^{1} x)}{36}$

Step 4.1.3.3.6

Raise $x$ to the power of $1$ .

$0 - \frac{- x^{2} + 12 x - 2 (x^{1} x^{1})}{36}$

Step 4.1.3.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$0 - \frac{- x^{2} + 12 x - 2 x^{1 + 1}}{36}$

Step 4.1.3.3.8

Add $1$ and $1$ .

$0 - \frac{- x^{2} + 12 x - 2 x^{2}}{36}$

Step 4.1.3.3.9

Subtract $2 x^{2}$ from $- x^{2}$ .

$0 - \frac{- 3 x^{2} + 12 x}{36}$

Step 4.1.3.3.10

Subtract $\frac{- 3 x^{2} + 12 x}{36}$ from $0$ .

$- \frac{- 3 x^{2} + 12 x}{36}$

Step 4.1.3.4

Factor $3 x$ out of $- 3 x^{2} + 12 x$ .

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Step 4.1.3.4.1

Factor $3 x$ out of $- 3 x^{2}$ .

$- \frac{3 x (- x) + 12 x}{36}$

Step 4.1.3.4.2

Factor $3 x$ out of $12 x$ .

$- \frac{3 x (- x) + 3 x (4)}{36}$

Step 4.1.3.4.3

Factor $3 x$ out of $3 x (- x) + 3 x (4)$ .

$- \frac{3 x (- x + 4)}{36}$

Step 4.1.3.5

Cancel the common factor of $3$ and $36$ .

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Step 4.1.3.5.1

Factor $3$ out of $3 x (- x + 4)$ .

$- \frac{3 (x (- x + 4))}{36}$

Step 4.1.3.5.2

Cancel the common factors.

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Step 4.1.3.5.2.1

Factor $3$ out of $36$ .

$- \frac{3 (x (- x + 4))}{3 \cdot 12}$

Step 4.1.3.5.2.2

Cancel the common factor.

$- \frac{3 (x (- x + 4))}{3 \cdot 12}$

Step 4.1.3.5.2.3

Rewrite the expression.

$- \frac{x (- x + 4)}{12}$

Step 4.1.3.6

Factor $- 1$ out of $- x$ .

$- \frac{x (- (x) + 4)}{12}$

Step 4.1.3.7

Rewrite $4$ as $- 1 (- 4)$ .

$- \frac{x (- (x) - 1 (- 4))}{12}$

Step 4.1.3.8

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$- \frac{x (- (x - 4))}{12}$

Step 4.1.3.9

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$- \frac{x (- 1 (x - 4))}{12}$

Step 4.1.3.10

Move the negative in front of the fraction.

$- - \frac{x (x - 4)}{12}$

Step 4.1.3.11

Multiply $- 1$ by $- 1$ .

$1 \frac{x (x - 4)}{12}$

Step 4.1.3.12

Multiply $\frac{x (x - 4)}{12}$ by $1$ .

$f' (x) = \frac{x (x - 4)}{12}$

Step 4.2

The first derivative of $t (x)$ with respect to $x$ is $\frac{x (x - 4)}{12}$ .

$\frac{x (x - 4)}{12}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{x (x - 4)}{12} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{x (x - 4)}{12} = 0$

Step 5.2

Set the numerator equal to zero.

$x (x - 4) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 4 = 0$

Step 5.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 5.3.3

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 5.3.3.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 5.3.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 5.3.4

The final solution is all the values that make $x (x - 4) = 0$ true.

$x = 0, 4$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 0, 4$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{0}{6} - \frac{1}{3}$

Step 9

Evaluate the second derivative.

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Step 9.1

Divide $0$ by $6$ .

$0 - \frac{1}{3}$

Step 9.2

Subtract $\frac{1}{3}$ from $0$ .

$- \frac{1}{3}$

Step 10

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 11

Find the y-value when $x = 0$ .

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Step 11.1

Replace the variable $x$ with $0$ in the expression.

$t (0) = 101 - \frac{1}{6} \cdot ({(0)}^{2} (1 - \frac{0}{6}))$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Raising $0$ to any positive power yields $0$ .

$t (0) = 101 - \frac{1}{6} \cdot (0 (1 - \frac{0}{6}))$

Step 11.2.1.2

Cancel the common factor of $6$ .

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Step 11.2.1.2.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$t (0) = 101 + \frac{- 1}{6} \cdot (0 (1 - \frac{0}{6}))$

Step 11.2.1.2.2

Factor $6$ out of $0 (1 - \frac{0}{6})$ .

$t (0) = 101 + \frac{- 1}{6} \cdot (6 (0 (1 - \frac{0}{6})))$

Step 11.2.1.2.3

Cancel the common factor.

$t (0) = 101 + \frac{- 1}{6} \cdot (6 (0 (1 - \frac{0}{6})))$

Step 11.2.1.2.4

Rewrite the expression.

$t (0) = 101 - 1 \cdot (0 (1 - \frac{0}{6}))$

Step 11.2.1.3

Multiply $0$ by $1 - \frac{0}{6}$ .

$t (0) = 101 - 1 \cdot 0$

Step 11.2.1.4

Multiply $- 1$ by $0$ .

$t (0) = 101 + 0$

Step 11.2.2

Add $101$ and $0$ .

$t (0) = 101$

Step 11.2.3

The final answer is $101$ .

$y = 101$

Step 12

Evaluate the second derivative at $x = 4$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4}{6} - \frac{1}{3}$

Step 13

Evaluate the second derivative.

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Step 13.1

Cancel the common factor of $4$ and $6$ .

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Step 13.1.1

Factor $2$ out of $4$ .

$\frac{2 (2)}{6} - \frac{1}{3}$

Step 13.1.2

Cancel the common factors.

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Step 13.1.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 2}{2 \cdot 3} - \frac{1}{3}$

Step 13.1.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 3} - \frac{1}{3}$

Step 13.1.2.3

Rewrite the expression.

$\frac{2}{3} - \frac{1}{3}$

Step 13.2

Combine the numerators over the common denominator.

$\frac{2 - 1}{3}$

Step 13.3

Subtract $1$ from $2$ .

$\frac{1}{3}$

Step 14

$x = 4$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 4$ is a local minimum

Step 15

Find the y-value when $x = 4$ .

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Step 15.1

Replace the variable $x$ with $4$ in the expression.

$t (4) = 101 - \frac{1}{6} \cdot ({(4)}^{2} (1 - \frac{4}{6}))$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

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

$t (4) = 101 - \frac{1}{6} \cdot (16 (1 - \frac{4}{6}))$

Step 15.2.1.2

Cancel the common factor of $2$ .

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Step 15.2.1.2.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$t (4) = 101 + \frac{- 1}{6} \cdot (16 (1 - \frac{4}{6}))$

Step 15.2.1.2.2

Factor $2$ out of $6$ .

$t (4) = 101 + \frac{- 1}{2 (3)} \cdot (16 (1 - \frac{4}{6}))$

Step 15.2.1.2.3

Factor $2$ out of $16 (1 - \frac{4}{6})$ .

$t (4) = 101 + \frac{- 1}{2 (3)} \cdot (2 (8 (1 - \frac{4}{6})))$

Step 15.2.1.2.4

Cancel the common factor.

$t (4) = 101 + \frac{- 1}{2 \cdot 3} \cdot (2 (8 (1 - \frac{4}{6})))$

Step 15.2.1.2.5

Rewrite the expression.

$t (4) = 101 + \frac{- 1}{3} \cdot (8 (1 - \frac{4}{6}))$

Step 15.2.1.3

Combine $8$ and $\frac{- 1}{3}$ .

$t (4) = 101 + \frac{8 \cdot - 1}{3} \cdot (1 - \frac{4}{6})$

Step 15.2.1.4

Multiply $8$ by $- 1$ .

$t (4) = 101 + \frac{- 8}{3} \cdot (1 - \frac{4}{6})$

Step 15.2.1.5

Move the negative in front of the fraction.

$t (4) = 101 - \frac{8}{3} \cdot (1 - \frac{4}{6})$

Step 15.2.1.6

Cancel the common factor of $4$ and $6$ .

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Step 15.2.1.6.1

Factor $2$ out of $4$ .

$t (4) = 101 - \frac{8}{3} \cdot (1 - \frac{2 (2)}{6})$

Step 15.2.1.6.2

Cancel the common factors.

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Step 15.2.1.6.2.1

Factor $2$ out of $6$ .

$t (4) = 101 - \frac{8}{3} \cdot (1 - \frac{2 \cdot 2}{2 \cdot 3})$

Step 15.2.1.6.2.2

Cancel the common factor.

$t (4) = 101 - \frac{8}{3} \cdot (1 - \frac{2 \cdot 2}{2 \cdot 3})$

Step 15.2.1.6.2.3

Rewrite the expression.

$t (4) = 101 - \frac{8}{3} \cdot (1 - \frac{2}{3})$

Step 15.2.1.7

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

$t (4) = 101 - \frac{8}{3} \cdot (\frac{3}{3} - \frac{2}{3})$

Step 15.2.1.8

Combine the numerators over the common denominator.

$t (4) = 101 - \frac{8}{3} \cdot \frac{3 - 2}{3}$

Step 15.2.1.9

Subtract $2$ from $3$ .

$t (4) = 101 - \frac{8}{3} \cdot \frac{1}{3}$

Step 15.2.1.10

Multiply $- \frac{8}{3} \cdot \frac{1}{3}$ .

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Step 15.2.1.10.1

Multiply $\frac{1}{3}$ by $\frac{8}{3}$ .

$t (4) = 101 - \frac{8}{3 \cdot 3}$

Step 15.2.1.10.2

Multiply $3$ by $3$ .

$t (4) = 101 - \frac{8}{9}$

Step 15.2.2

To write $101$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$t (4) = 101 \cdot \frac{9}{9} - \frac{8}{9}$

Step 15.2.3

Combine $101$ and $\frac{9}{9}$ .

$t (4) = \frac{101 \cdot 9}{9} - \frac{8}{9}$

Step 15.2.4

Combine the numerators over the common denominator.

$t (4) = \frac{101 \cdot 9 - 8}{9}$

Step 15.2.5

Simplify the numerator.

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Step 15.2.5.1

Multiply $101$ by $9$ .

$t (4) = \frac{909 - 8}{9}$

Step 15.2.5.2

Subtract $8$ from $909$ .

$t (4) = \frac{901}{9}$

Step 15.2.6

The final answer is $\frac{901}{9}$ .

$y = \frac{901}{9}$

Step 16

These are the local extrema for $t (x) = 101 - \frac{1}{6} \cdot (x^{2} (1 - \frac{x}{6}))$ .

$(0, 101)$ is a local maxima

$(4, \frac{901}{9})$ is a local minima

Step 17