Calculus Examples

$f (x) = 136 x^{7} + 576 x^{7} \ln (3 x)$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $136 x^{7} + 576 x^{7} \ln (3 x)$ with respect to $x$ is $\frac{d}{d x} [136 x^{7}] + \frac{d}{d x} [576 x^{7} \ln (3 x)]$ .

$\frac{d}{d x} [136 x^{7}] + \frac{d}{d x} [576 x^{7} \ln (3 x)]$

Step 1.2

Evaluate $\frac{d}{d x} [136 x^{7}]$ .

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Step 1.2.1

Since $136$ is constant with respect to $x$ , the derivative of $136 x^{7}$ with respect to $x$ is $136 \frac{d}{d x} [x^{7}]$ .

$136 \frac{d}{d x} [x^{7}] + \frac{d}{d x} [576 x^{7} \ln (3 x)]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$136 (7 x^{6}) + \frac{d}{d x} [576 x^{7} \ln (3 x)]$

Step 1.2.3

Multiply $7$ by $136$ .

$952 x^{6} + \frac{d}{d x} [576 x^{7} \ln (3 x)]$

Step 1.3

Evaluate $\frac{d}{d x} [576 x^{7} \ln (3 x)]$ .

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Step 1.3.1

Since $576$ is constant with respect to $x$ , the derivative of $576 x^{7} \ln (3 x)$ with respect to $x$ is $576 \frac{d}{d x} [x^{7} \ln (3 x)]$ .

$952 x^{6} + 576 \frac{d}{d x} [x^{7} \ln (3 x)]$

Step 1.3.2

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 $f (x) = x^{7}$ and $g (x) = \ln (3 x)$ .

$952 x^{6} + 576 (x^{7} \frac{d}{d x} [\ln (3 x)] + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 x$ .

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Step 1.3.3.1

To apply the Chain Rule, set $u$ as $3 x$ .

$952 x^{6} + 576 (x^{7} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x]) + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 1.3.3.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$952 x^{6} + 576 (x^{7} (\frac{1}{u} \frac{d}{d x} [3 x]) + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 1.3.3.3

Replace all occurrences of $u$ with $3 x$ .

$952 x^{6} + 576 (x^{7} (\frac{1}{3 x} \frac{d}{d x} [3 x]) + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 1.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$952 x^{6} + 576 (x^{7} (\frac{1}{3 x} (3 \frac{d}{d x} [x])) + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 1.3.5

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

$952 x^{6} + 576 (x^{7} (\frac{1}{3 x} (3 \cdot 1)) + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$952 x^{6} + 576 (x^{7} (\frac{1}{3 x} (3 \cdot 1)) + \ln (3 x) (7 x^{6}))$

Step 1.3.7

Multiply $3$ by $1$ .

$952 x^{6} + 576 (x^{7} (\frac{1}{3 x} \cdot 3) + \ln (3 x) (7 x^{6}))$

Step 1.3.8

Combine $\frac{1}{3 x}$ and $3$ .

$952 x^{6} + 576 (x^{7} \frac{3}{3 x} + \ln (3 x) (7 x^{6}))$

Step 1.3.9

Cancel the common factor of $3$ .

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Step 1.3.9.1

Cancel the common factor.

$952 x^{6} + 576 (x^{7} \frac{3}{3 x} + \ln (3 x) (7 x^{6}))$

Step 1.3.9.2

Rewrite the expression.

$952 x^{6} + 576 (x^{7} \frac{1}{x} + \ln (3 x) (7 x^{6}))$

Step 1.3.10

Combine $x^{7}$ and $\frac{1}{x}$ .

$952 x^{6} + 576 (\frac{x^{7}}{x} + \ln (3 x) (7 x^{6}))$

Step 1.3.11

Cancel the common factor of $x^{7}$ and $x$ .

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Step 1.3.11.1

Factor $x$ out of $x^{7}$ .

$952 x^{6} + 576 (\frac{x \cdot x^{6}}{x} + \ln (3 x) (7 x^{6}))$

Step 1.3.11.2

Cancel the common factors.

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Step 1.3.11.2.1

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

$952 x^{6} + 576 (\frac{x \cdot x^{6}}{x^{1}} + \ln (3 x) (7 x^{6}))$

Step 1.3.11.2.2

Factor $x$ out of $x^{1}$ .

$952 x^{6} + 576 (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (3 x) (7 x^{6}))$

Step 1.3.11.2.3

Cancel the common factor.

$952 x^{6} + 576 (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (3 x) (7 x^{6}))$

Step 1.3.11.2.4

Rewrite the expression.

$952 x^{6} + 576 (\frac{x^{6}}{1} + \ln (3 x) (7 x^{6}))$

Step 1.3.11.2.5

Divide $x^{6}$ by $1$ .

$952 x^{6} + 576 (x^{6} + \ln (3 x) (7 x^{6}))$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$952 x^{6} + 576 x^{6} + 576 (\ln (3 x) (7 x^{6}))$

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Multiply $7$ by $576$ .

$952 x^{6} + 576 x^{6} + 4032 (\ln (3 x) (x^{6}))$

Step 1.4.2.2

Add $952 x^{6}$ and $576 x^{6}$ .

$1528 x^{6} + 4032 \ln (3 x) x^{6}$

Step 1.4.3

Reorder terms.

$1528 x^{6} + 4032 x^{6} \ln (3 x)$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $1528 x^{6} + 4032 x^{6} \ln (3 x)$ with respect to $x$ is $\frac{d}{d x} [1528 x^{6}] + \frac{d}{d x} [4032 x^{6} \ln (3 x)]$ .

$f'' (x) = \frac{d}{d x} (1528 x^{6}) + \frac{d}{d x} (4032 x^{6} \ln (3 x))$

Step 2.2

Evaluate $\frac{d}{d x} [1528 x^{6}]$ .

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Step 2.2.1

Since $1528$ is constant with respect to $x$ , the derivative of $1528 x^{6}$ with respect to $x$ is $1528 \frac{d}{d x} [x^{6}]$ .

$f'' (x) = 1528 \frac{d}{d x} (x^{6}) + \frac{d}{d x} (4032 x^{6} \ln (3 x))$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$f'' (x) = 1528 (6 x^{5}) + \frac{d}{d x} (4032 x^{6} \ln (3 x))$

Step 2.2.3

Multiply $6$ by $1528$ .

$f'' (x) = 9168 x^{5} + \frac{d}{d x} (4032 x^{6} \ln (3 x))$

Step 2.3

Evaluate $\frac{d}{d x} [4032 x^{6} \ln (3 x)]$ .

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Step 2.3.1

Since $4032$ is constant with respect to $x$ , the derivative of $4032 x^{6} \ln (3 x)$ with respect to $x$ is $4032 \frac{d}{d x} [x^{6} \ln (3 x)]$ .

$f'' (x) = 9168 x^{5} + 4032 \frac{d}{d x} (x^{6} \ln (3 x))$

Step 2.3.2

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 $f (x) = x^{6}$ and $g (x) = \ln (3 x)$ .

$f'' (x) = 9168 x^{5} + 4032 (x^{6} \frac{d}{d x} (\ln (3 x)) + \ln (3 x) \frac{d}{d x} (x^{6}))$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 x$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $3 x$ .

$f'' (x) = 9168 x^{5} + 4032 (x^{6} (\frac{d}{d u} (\ln (u)) \frac{d}{d x} (3 x)) + \ln (3 x) \frac{d}{d x} (x^{6}))$

Step 2.3.3.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$f'' (x) = 9168 x^{5} + 4032 (x^{6} (\frac{1}{u} \cdot \frac{d}{d x} (3 x)) + \ln (3 x) \frac{d}{d x} (x^{6}))$

Step 2.3.3.3

Replace all occurrences of $u$ with $3 x$ .

$f'' (x) = 9168 x^{5} + 4032 (x^{6} (\frac{1}{3 x} \cdot \frac{d}{d x} (3 x)) + \ln (3 x) \frac{d}{d x} (x^{6}))$

Step 2.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$f'' (x) = 9168 x^{5} + 4032 (x^{6} (\frac{1}{3 x} \cdot (3 \frac{d}{d x} (x))) + \ln (3 x) \frac{d}{d x} (x^{6}))$

Step 2.3.5

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

$f'' (x) = 9168 x^{5} + 4032 (x^{6} (\frac{1}{3 x} \cdot (3 \cdot 1)) + \ln (3 x) \frac{d}{d x} (x^{6}))$

Step 2.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$f'' (x) = 9168 x^{5} + 4032 (x^{6} (\frac{1}{3 x} \cdot (3 \cdot 1)) + \ln (3 x) (6 x^{5}))$

Step 2.3.7

Multiply $3$ by $1$ .

$f'' (x) = 9168 x^{5} + 4032 (x^{6} (\frac{1}{3 x} \cdot 3) + \ln (3 x) (6 x^{5}))$

Step 2.3.8

Combine $\frac{1}{3 x}$ and $3$ .

$f'' (x) = 9168 x^{5} + 4032 (x^{6} (\frac{3}{3 x}) + \ln (3 x) (6 x^{5}))$

Step 2.3.9

Cancel the common factor of $3$ .

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Step 2.3.9.1

Cancel the common factor.

$f'' (x) = 9168 x^{5} + 4032 (x^{6} (\frac{3}{3 x}) + \ln (3 x) (6 x^{5}))$

Step 2.3.9.2

Rewrite the expression.

$f'' (x) = 9168 x^{5} + 4032 (x^{6} (\frac{1}{x}) + \ln (3 x) (6 x^{5}))$

Step 2.3.10

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

$f'' (x) = 9168 x^{5} + 4032 (\frac{x^{6}}{x} + \ln (3 x) (6 x^{5}))$

Step 2.3.11

Cancel the common factor of $x^{6}$ and $x$ .

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Step 2.3.11.1

Factor $x$ out of $x^{6}$ .

$f'' (x) = 9168 x^{5} + 4032 (\frac{x \cdot x^{5}}{x} + \ln (3 x) (6 x^{5}))$

Step 2.3.11.2

Cancel the common factors.

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Step 2.3.11.2.1

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

$f'' (x) = 9168 x^{5} + 4032 (\frac{x \cdot x^{5}}{x} + \ln (3 x) (6 x^{5}))$

Step 2.3.11.2.2

Factor $x$ out of $x^{1}$ .

$f'' (x) = 9168 x^{5} + 4032 (\frac{x \cdot x^{5}}{x \cdot 1} + \ln (3 x) (6 x^{5}))$

Step 2.3.11.2.3

Cancel the common factor.

$f'' (x) = 9168 x^{5} + 4032 (\frac{x \cdot x^{5}}{x \cdot 1} + \ln (3 x) (6 x^{5}))$

Step 2.3.11.2.4

Rewrite the expression.

$f'' (x) = 9168 x^{5} + 4032 (\frac{x^{5}}{1} + \ln (3 x) (6 x^{5}))$

Step 2.3.11.2.5

Divide $x^{5}$ by $1$ .

$f'' (x) = 9168 x^{5} + 4032 (x^{5} + \ln (3 x) (6 x^{5}))$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$f'' (x) = 9168 x^{5} + 4032 x^{5} + 4032 (\ln (3 x) (6 x^{5}))$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $6$ by $4032$ .

$f'' (x) = 9168 x^{5} + 4032 x^{5} + 24192 (\ln (3 x) (x^{5}))$

Step 2.4.2.2

Add $9168 x^{5}$ and $4032 x^{5}$ .

$f'' (x) = 13200 x^{5} + 24192 \ln (3 x) x^{5}$

Step 2.4.3

Reorder terms.

$f'' (x) = 13200 x^{5} + 24192 x^{5} \ln (3 x)$

Step 3

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

$1528 x^{6} + 4032 x^{6} \ln (3 x) = 0$

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

By the Sum Rule, the derivative of $136 x^{7} + 576 x^{7} \ln (3 x)$ with respect to $x$ is $\frac{d}{d x} [136 x^{7}] + \frac{d}{d x} [576 x^{7} \ln (3 x)]$ .

$\frac{d}{d x} [136 x^{7}] + \frac{d}{d x} [576 x^{7} \ln (3 x)]$

Step 4.1.2

Evaluate $\frac{d}{d x} [136 x^{7}]$ .

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Step 4.1.2.1

Since $136$ is constant with respect to $x$ , the derivative of $136 x^{7}$ with respect to $x$ is $136 \frac{d}{d x} [x^{7}]$ .

$136 \frac{d}{d x} [x^{7}] + \frac{d}{d x} [576 x^{7} \ln (3 x)]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$136 (7 x^{6}) + \frac{d}{d x} [576 x^{7} \ln (3 x)]$

Step 4.1.2.3

Multiply $7$ by $136$ .

$952 x^{6} + \frac{d}{d x} [576 x^{7} \ln (3 x)]$

Step 4.1.3

Evaluate $\frac{d}{d x} [576 x^{7} \ln (3 x)]$ .

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Step 4.1.3.1

Since $576$ is constant with respect to $x$ , the derivative of $576 x^{7} \ln (3 x)$ with respect to $x$ is $576 \frac{d}{d x} [x^{7} \ln (3 x)]$ .

$952 x^{6} + 576 \frac{d}{d x} [x^{7} \ln (3 x)]$

Step 4.1.3.2

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 $f (x) = x^{7}$ and $g (x) = \ln (3 x)$ .

$952 x^{6} + 576 (x^{7} \frac{d}{d x} [\ln (3 x)] + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 4.1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 x$ .

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Step 4.1.3.3.1

To apply the Chain Rule, set $u$ as $3 x$ .

$952 x^{6} + 576 (x^{7} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x]) + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 4.1.3.3.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$952 x^{6} + 576 (x^{7} (\frac{1}{u} \frac{d}{d x} [3 x]) + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 4.1.3.3.3

Replace all occurrences of $u$ with $3 x$ .

$952 x^{6} + 576 (x^{7} (\frac{1}{3 x} \frac{d}{d x} [3 x]) + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 4.1.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$952 x^{6} + 576 (x^{7} (\frac{1}{3 x} (3 \frac{d}{d x} [x])) + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 4.1.3.5

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

$952 x^{6} + 576 (x^{7} (\frac{1}{3 x} (3 \cdot 1)) + \ln (3 x) \frac{d}{d x} [x^{7}])$

Step 4.1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$952 x^{6} + 576 (x^{7} (\frac{1}{3 x} (3 \cdot 1)) + \ln (3 x) (7 x^{6}))$

Step 4.1.3.7

Multiply $3$ by $1$ .

$952 x^{6} + 576 (x^{7} (\frac{1}{3 x} \cdot 3) + \ln (3 x) (7 x^{6}))$

Step 4.1.3.8

Combine $\frac{1}{3 x}$ and $3$ .

$952 x^{6} + 576 (x^{7} \frac{3}{3 x} + \ln (3 x) (7 x^{6}))$

Step 4.1.3.9

Cancel the common factor of $3$ .

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Step 4.1.3.9.1

Cancel the common factor.

$952 x^{6} + 576 (x^{7} \frac{3}{3 x} + \ln (3 x) (7 x^{6}))$

Step 4.1.3.9.2

Rewrite the expression.

$952 x^{6} + 576 (x^{7} \frac{1}{x} + \ln (3 x) (7 x^{6}))$

Step 4.1.3.10

Combine $x^{7}$ and $\frac{1}{x}$ .

$952 x^{6} + 576 (\frac{x^{7}}{x} + \ln (3 x) (7 x^{6}))$

Step 4.1.3.11

Cancel the common factor of $x^{7}$ and $x$ .

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Step 4.1.3.11.1

Factor $x$ out of $x^{7}$ .

$952 x^{6} + 576 (\frac{x \cdot x^{6}}{x} + \ln (3 x) (7 x^{6}))$

Step 4.1.3.11.2

Cancel the common factors.

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Step 4.1.3.11.2.1

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

$952 x^{6} + 576 (\frac{x \cdot x^{6}}{x^{1}} + \ln (3 x) (7 x^{6}))$

Step 4.1.3.11.2.2

Factor $x$ out of $x^{1}$ .

$952 x^{6} + 576 (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (3 x) (7 x^{6}))$

Step 4.1.3.11.2.3

Cancel the common factor.

$952 x^{6} + 576 (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (3 x) (7 x^{6}))$

Step 4.1.3.11.2.4

Rewrite the expression.

$952 x^{6} + 576 (\frac{x^{6}}{1} + \ln (3 x) (7 x^{6}))$

Step 4.1.3.11.2.5

Divide $x^{6}$ by $1$ .

$952 x^{6} + 576 (x^{6} + \ln (3 x) (7 x^{6}))$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Apply the distributive property.

$952 x^{6} + 576 x^{6} + 576 (\ln (3 x) (7 x^{6}))$

Step 4.1.4.2

Combine terms.

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Step 4.1.4.2.1

Multiply $7$ by $576$ .

$952 x^{6} + 576 x^{6} + 4032 (\ln (3 x) (x^{6}))$

Step 4.1.4.2.2

Add $952 x^{6}$ and $576 x^{6}$ .

$1528 x^{6} + 4032 \ln (3 x) x^{6}$

Step 4.1.4.3

Reorder terms.

$f' (x) = 1528 x^{6} + 4032 x^{6} \ln (3 x)$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $1528 x^{6} + 4032 x^{6} \ln (3 x)$ .

$1528 x^{6} + 4032 x^{6} \ln (3 x)$

Step 5

Set the first derivative equal to $0$ then solve the equation $1528 x^{6} + 4032 x^{6} \ln (3 x) = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$1528 x^{6} + 4032 x^{6} \ln (3 x) = 0$

Step 5.2

Subtract $1528 x^{6}$ from both sides of the equation.

$4032 x^{6} \ln (3 x) = - 1528 x^{6}$

Step 5.3

Divide each term in $4032 x^{6} \ln (3 x) = - 1528 x^{6}$ by $4032 x^{6}$ and simplify.

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Step 5.3.1

Divide each term in $4032 x^{6} \ln (3 x) = - 1528 x^{6}$ by $4032 x^{6}$ .

$\frac{4032 x^{6} \ln (3 x)}{4032 x^{6}} = \frac{- 1528 x^{6}}{4032 x^{6}}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $4032$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{4032 x^{6} \ln (3 x)}{4032 x^{6}} = \frac{- 1528 x^{6}}{4032 x^{6}}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{x^{6} \ln (3 x)}{x^{6}} = \frac{- 1528 x^{6}}{4032 x^{6}}$

Step 5.3.2.2

Cancel the common factor of $x^{6}$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{x^{6} \ln (3 x)}{x^{6}} = \frac{- 1528 x^{6}}{4032 x^{6}}$

Step 5.3.2.2.2

Divide $\ln (3 x)$ by $1$ .

$\ln (3 x) = \frac{- 1528 x^{6}}{4032 x^{6}}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $- 1528$ and $4032$ .

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Step 5.3.3.1.1

Factor $8$ out of $- 1528 x^{6}$ .

$\ln (3 x) = \frac{8 (- 191 x^{6})}{4032 x^{6}}$

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

Factor $8$ out of $4032 x^{6}$ .

$\ln (3 x) = \frac{8 (- 191 x^{6})}{8 (504 x^{6})}$

Step 5.3.3.1.2.2

Cancel the common factor.

$\ln (3 x) = \frac{8 (- 191 x^{6})}{8 (504 x^{6})}$

Step 5.3.3.1.2.3

Rewrite the expression.

$\ln (3 x) = \frac{- 191 x^{6}}{504 x^{6}}$

Step 5.3.3.2

Cancel the common factor of $x^{6}$ .

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Step 5.3.3.2.1

Cancel the common factor.

$\ln (3 x) = \frac{- 191 x^{6}}{504 x^{6}}$

Step 5.3.3.2.2

Rewrite the expression.

$\ln (3 x) = \frac{- 191}{504}$

Step 5.3.3.3

Move the negative in front of the fraction.

$\ln (3 x) = - \frac{191}{504}$

Step 5.4

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (3 x)} = e^{- \frac{191}{504}}$

Step 5.5

Rewrite $\ln (3 x) = - \frac{191}{504}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- \frac{191}{504}} = 3 x$

Step 5.6

Solve for $x$ .

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Step 5.6.1

Rewrite the equation as $3 x = e^{- \frac{191}{504}}$ .

$3 x = e^{- \frac{191}{504}}$

Step 5.6.2

Divide each term in $3 x = e^{- \frac{191}{504}}$ by $3$ and simplify.

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Step 5.6.2.1

Divide each term in $3 x = e^{- \frac{191}{504}}$ by $3$ .

$\frac{3 x}{3} = \frac{e^{- \frac{191}{504}}}{3}$

Step 5.6.2.2

Simplify the left side.

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Step 5.6.2.2.1

Cancel the common factor of $3$ .

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Step 5.6.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{e^{- \frac{191}{504}}}{3}$

Step 5.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{- \frac{191}{504}}}{3}$

Step 5.6.2.3

Simplify the right side.

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Step 5.6.2.3.1

Move $e^{- \frac{191}{504}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{3 e^{\frac{191}{504}}}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the argument in $\ln (3 x)$ less than or equal to $0$ to find where the expression is undefined.

$3 x \leq 0$

Step 6.2

Divide each term in $3 x \leq 0$ by $3$ and simplify.

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Step 6.2.1

Divide each term in $3 x \leq 0$ by $3$ .

$\frac{3 x}{3} \leq \frac{0}{3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \leq \frac{0}{3}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{0}{3}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $0$ by $3$ .

$x \leq 0$

Step 6.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 7

Critical points to evaluate.

$x = \frac{1}{3 e^{\frac{191}{504}}}$

Step 8

Evaluate the second derivative at $x = \frac{1}{3 e^{\frac{191}{504}}}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$13200 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.1.1.1

Apply the product rule to $\frac{1}{3 e^{\frac{191}{504}}}$ .

$13200 \frac{1^{5}}{{(3 e^{\frac{191}{504}})}^{5}} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.1.2

Apply the product rule to $3 e^{\frac{191}{504}}$ .

$13200 \frac{1^{5}}{3^{5} {(e^{\frac{191}{504}})}^{5}} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.2

One to any power is one.

$13200 \frac{1}{3^{5} {(e^{\frac{191}{504}})}^{5}} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.3

Simplify the denominator.

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Step 9.1.3.1

Raise $3$ to the power of $5$ .

$13200 \frac{1}{243 {(e^{\frac{191}{504}})}^{5}} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.3.2

Multiply the exponents in ${(e^{\frac{191}{504}})}^{5}$ .

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Step 9.1.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$13200 \frac{1}{243 e^{\frac{191}{504} \cdot 5}} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.3.2.2

Multiply $\frac{191}{504} \cdot 5$ .

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Step 9.1.3.2.2.1

Combine $\frac{191}{504}$ and $5$ .

$13200 \frac{1}{243 e^{\frac{191 \cdot 5}{504}}} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.3.2.2.2

Multiply $191$ by $5$ .

$13200 \frac{1}{243 e^{\frac{955}{504}}} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.4

Cancel the common factor of $3$ .

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Step 9.1.4.1

Factor $3$ out of $13200$ .

$3 (4400) \frac{1}{243 e^{\frac{955}{504}}} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.4.2

Factor $3$ out of $243 e^{\frac{955}{504}}$ .

$3 (4400) \frac{1}{3 (81 e^{\frac{955}{504}})} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.4.3

Cancel the common factor.

$3 \cdot 4400 \frac{1}{3 (81 e^{\frac{955}{504}})} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.4.4

Rewrite the expression.

$4400 \frac{1}{81 e^{\frac{955}{504}}} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.5

Combine $4400$ and $\frac{1}{81 e^{\frac{955}{504}}}$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + 24192 {(\frac{1}{3 e^{\frac{191}{504}}})}^{5} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.1.6.1

Apply the product rule to $\frac{1}{3 e^{\frac{191}{504}}}$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + 24192 \frac{1^{5}}{{(3 e^{\frac{191}{504}})}^{5}} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.6.2

Apply the product rule to $3 e^{\frac{191}{504}}$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + 24192 \frac{1^{5}}{3^{5} {(e^{\frac{191}{504}})}^{5}} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.7

One to any power is one.

$\frac{4400}{81 e^{\frac{955}{504}}} + 24192 \frac{1}{3^{5} {(e^{\frac{191}{504}})}^{5}} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.8

Simplify the denominator.

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Step 9.1.8.1

Raise $3$ to the power of $5$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + 24192 \frac{1}{243 {(e^{\frac{191}{504}})}^{5}} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.8.2

Multiply the exponents in ${(e^{\frac{191}{504}})}^{5}$ .

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Step 9.1.8.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + 24192 \frac{1}{243 e^{\frac{191}{504} \cdot 5}} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.8.2.2

Multiply $\frac{191}{504} \cdot 5$ .

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Step 9.1.8.2.2.1

Combine $\frac{191}{504}$ and $5$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + 24192 \frac{1}{243 e^{\frac{191 \cdot 5}{504}}} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.8.2.2.2

Multiply $191$ by $5$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + 24192 \frac{1}{243 e^{\frac{955}{504}}} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.9

Cancel the common factor of $27$ .

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Step 9.1.9.1

Factor $27$ out of $24192$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + 27 (896) \frac{1}{243 e^{\frac{955}{504}}} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.9.2

Factor $27$ out of $243 e^{\frac{955}{504}}$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + 27 (896) \frac{1}{27 (9 e^{\frac{955}{504}})} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.9.3

Cancel the common factor.

$\frac{4400}{81 e^{\frac{955}{504}}} + 27 \cdot 896 \frac{1}{27 (9 e^{\frac{955}{504}})} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.9.4

Rewrite the expression.

$\frac{4400}{81 e^{\frac{955}{504}}} + 896 \frac{1}{9 e^{\frac{955}{504}}} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.10

Combine $896$ and $\frac{1}{9 e^{\frac{955}{504}}}$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{896}{9 e^{\frac{955}{504}}} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 9.1.11

Cancel the common factor of $3$ .

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Step 9.1.11.1

Cancel the common factor.

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{896}{9 e^{\frac{955}{504}}} \ln (3 \frac{1}{3 e^{\frac{191}{504}}})$

Step 9.1.11.2

Rewrite the expression.

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{896}{9 e^{\frac{955}{504}}} \ln (\frac{1}{e^{\frac{191}{504}}})$

Step 9.1.12

Move $e^{\frac{191}{504}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{896}{9 e^{\frac{955}{504}}} \ln (e^{- \frac{191}{504}})$

Step 9.1.13

Expand $\ln (e^{- \frac{191}{504}})$ by moving $- \frac{191}{504}$ outside the logarithm.

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{896}{9 e^{\frac{955}{504}}} (- \frac{191}{504} \ln (e))$

Step 9.1.14

The natural logarithm of $e$ is $1$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{896}{9 e^{\frac{955}{504}}} (- \frac{191}{504} \cdot 1)$

Step 9.1.15

Multiply $- 1$ by $1$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{896}{9 e^{\frac{955}{504}}} (- \frac{191}{504})$

Step 9.1.16

Cancel the common factor of $56$ .

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Step 9.1.16.1

Move the leading negative in $- \frac{191}{504}$ into the numerator.

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{896}{9 e^{\frac{955}{504}}} \cdot \frac{- 191}{504}$

Step 9.1.16.2

Factor $56$ out of $896$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{56 (16)}{9 e^{\frac{955}{504}}} \cdot \frac{- 191}{504}$

Step 9.1.16.3

Factor $56$ out of $504$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{56 \cdot 16}{9 e^{\frac{955}{504}}} \cdot \frac{- 191}{56 \cdot 9}$

Step 9.1.16.4

Cancel the common factor.

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{56 \cdot 16}{9 e^{\frac{955}{504}}} \cdot \frac{- 191}{56 \cdot 9}$

Step 9.1.16.5

Rewrite the expression.

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{16}{9 e^{\frac{955}{504}}} \cdot \frac{- 191}{9}$

Step 9.1.17

Multiply $\frac{16}{9 e^{\frac{955}{504}}}$ by $\frac{- 191}{9}$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{16 \cdot - 191}{9 e^{\frac{955}{504}} \cdot 9}$

Step 9.1.18

Multiply $16$ by $- 191$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{- 3056}{9 e^{\frac{955}{504}} \cdot 9}$

Step 9.1.19

Multiply $9$ by $9$ .

$\frac{4400}{81 e^{\frac{955}{504}}} + \frac{- 3056}{81 e^{\frac{955}{504}}}$

Step 9.1.20

Move the negative in front of the fraction.

$\frac{4400}{81 e^{\frac{955}{504}}} - \frac{3056}{81 e^{\frac{955}{504}}}$

Step 9.2

Simplify terms.

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Step 9.2.1

Combine the numerators over the common denominator.

$\frac{4400 - 3056}{81 e^{\frac{955}{504}}}$

Step 9.2.2

Subtract $3056$ from $4400$ .

$\frac{1344}{81 e^{\frac{955}{504}}}$

Step 9.2.3

Factor $3$ out of $1344$ .

$\frac{3 (448)}{81 e^{\frac{955}{504}}}$

Step 9.3

Cancel the common factors.

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Step 9.3.1

Factor $3$ out of $81 e^{\frac{955}{504}}$ .

$\frac{3 (448)}{3 (27 e^{\frac{955}{504}})}$

Step 9.3.2

Cancel the common factor.

$\frac{3 \cdot 448}{3 (27 e^{\frac{955}{504}})}$

Step 9.3.3

Rewrite the expression.

$\frac{448}{27 e^{\frac{955}{504}}}$

Step 10

$x = \frac{1}{3 e^{\frac{191}{504}}}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{1}{3 e^{\frac{191}{504}}}$ is a local minimum

Step 11

Find the y-value when $x = \frac{1}{3 e^{\frac{191}{504}}}$ .

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Step 11.1

Replace the variable $x$ with $\frac{1}{3 e^{\frac{191}{504}}}$ in the expression.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = 136 {(\frac{1}{3 e^{\frac{191}{504}}})}^{7} + 576 {(\frac{1}{3 e^{\frac{191}{504}}})}^{7} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.2.1.1.1

Apply the product rule to $\frac{1}{3 e^{\frac{191}{504}}}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = 136 (\frac{1^{7}}{{(3 e^{\frac{191}{504}})}^{7}}) + 576 {(\frac{1}{3 e^{\frac{191}{504}}})}^{7} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.1.2

Apply the product rule to $3 e^{\frac{191}{504}}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = 136 (\frac{1^{7}}{3^{7} {(e^{\frac{191}{504}})}^{7}}) + 576 {(\frac{1}{3 e^{\frac{191}{504}}})}^{7} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.2

One to any power is one.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = 136 (\frac{1}{3^{7} {(e^{\frac{191}{504}})}^{7}}) + 576 {(\frac{1}{3 e^{\frac{191}{504}}})}^{7} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.3

Simplify the denominator.

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Step 11.2.1.3.1

Raise $3$ to the power of $7$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = 136 (\frac{1}{2187 {(e^{\frac{191}{504}})}^{7}}) + 576 {(\frac{1}{3 e^{\frac{191}{504}}})}^{7} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.3.2

Multiply the exponents in ${(e^{\frac{191}{504}})}^{7}$ .

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Step 11.2.1.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = 136 (\frac{1}{2187 e^{\frac{191}{504} \cdot 7}}) + 576 {(\frac{1}{3 e^{\frac{191}{504}}})}^{7} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.3.2.2

Cancel the common factor of $7$ .

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Step 11.2.1.3.2.2.1

Factor $7$ out of $504$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = 136 (\frac{1}{2187 e^{\frac{191}{7 (72)} \cdot 7}}) + 576 {(\frac{1}{3 e^{\frac{191}{504}}})}^{7} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.3.2.2.2

Cancel the common factor.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = 136 (\frac{1}{2187 e^{\frac{191}{7 \cdot 72} \cdot 7}}) + 576 {(\frac{1}{3 e^{\frac{191}{504}}})}^{7} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.3.2.2.3

Rewrite the expression.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = 136 (\frac{1}{2187 e^{\frac{191}{72}}}) + 576 {(\frac{1}{3 e^{\frac{191}{504}}})}^{7} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.4

Combine $136$ and $\frac{1}{2187 e^{\frac{191}{72}}}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 576 {(\frac{1}{3 e^{\frac{191}{504}}})}^{7} \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.2.1.5.1

Apply the product rule to $\frac{1}{3 e^{\frac{191}{504}}}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 576 (\frac{1^{7}}{{(3 e^{\frac{191}{504}})}^{7}}) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.5.2

Apply the product rule to $3 e^{\frac{191}{504}}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 576 (\frac{1^{7}}{3^{7} {(e^{\frac{191}{504}})}^{7}}) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.6

One to any power is one.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 576 (\frac{1}{3^{7} {(e^{\frac{191}{504}})}^{7}}) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.7

Simplify the denominator.

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Step 11.2.1.7.1

Raise $3$ to the power of $7$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 576 (\frac{1}{2187 {(e^{\frac{191}{504}})}^{7}}) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.7.2

Multiply the exponents in ${(e^{\frac{191}{504}})}^{7}$ .

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Step 11.2.1.7.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 576 (\frac{1}{2187 e^{\frac{191}{504} \cdot 7}}) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.7.2.2

Cancel the common factor of $7$ .

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Step 11.2.1.7.2.2.1

Factor $7$ out of $504$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 576 (\frac{1}{2187 e^{\frac{191}{7 (72)} \cdot 7}}) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.7.2.2.2

Cancel the common factor.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 576 (\frac{1}{2187 e^{\frac{191}{7 \cdot 72} \cdot 7}}) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.7.2.2.3

Rewrite the expression.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 576 (\frac{1}{2187 e^{\frac{191}{72}}}) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.8

Cancel the common factor of $9$ .

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Step 11.2.1.8.1

Factor $9$ out of $576$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 9 (64) (\frac{1}{2187 e^{\frac{191}{72}}}) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.8.2

Factor $9$ out of $2187 e^{\frac{191}{72}}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 9 (64) (\frac{1}{9 (243 e^{\frac{191}{72}})}) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.8.3

Cancel the common factor.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 9 \cdot (64 (\frac{1}{9 (243 e^{\frac{191}{72}})})) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.8.4

Rewrite the expression.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + 64 (\frac{1}{243 e^{\frac{191}{72}}}) \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.9

Combine $64$ and $\frac{1}{243 e^{\frac{191}{72}}}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{64}{243 e^{\frac{191}{72}}} \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.10

Cancel the common factor of $3$ .

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Step 11.2.1.10.1

Cancel the common factor.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{64}{243 e^{\frac{191}{72}}} \cdot \ln (3 (\frac{1}{3 e^{\frac{191}{504}}}))$

Step 11.2.1.10.2

Rewrite the expression.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{64}{243 e^{\frac{191}{72}}} \cdot \ln (\frac{1}{e^{\frac{191}{504}}})$

Step 11.2.1.11

Move $e^{\frac{191}{504}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{64}{243 e^{\frac{191}{72}}} \cdot \ln (e^{- \frac{191}{504}})$

Step 11.2.1.12

Expand $\ln (e^{- \frac{191}{504}})$ by moving $- \frac{191}{504}$ outside the logarithm.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{64}{243 e^{\frac{191}{72}}} \cdot (- \frac{191}{504} \cdot \ln (e))$

Step 11.2.1.13

The natural logarithm of $e$ is $1$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{64}{243 e^{\frac{191}{72}}} \cdot (- \frac{191}{504} \cdot 1)$

Step 11.2.1.14

Multiply $- 1$ by $1$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{64}{243 e^{\frac{191}{72}}} \cdot (- \frac{191}{504})$

Step 11.2.1.15

Cancel the common factor of $8$ .

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Step 11.2.1.15.1

Move the leading negative in $- \frac{191}{504}$ into the numerator.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{64}{243 e^{\frac{191}{72}}} \cdot \frac{- 191}{504}$

Step 11.2.1.15.2

Factor $8$ out of $64$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{8 (8)}{243 e^{\frac{191}{72}}} \cdot \frac{- 191}{504}$

Step 11.2.1.15.3

Factor $8$ out of $504$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{8 \cdot 8}{243 e^{\frac{191}{72}}} \cdot \frac{- 191}{8 \cdot 63}$

Step 11.2.1.15.4

Cancel the common factor.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{8 \cdot 8}{243 e^{\frac{191}{72}}} \cdot \frac{- 191}{8 \cdot 63}$

Step 11.2.1.15.5

Rewrite the expression.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{8}{243 e^{\frac{191}{72}}} \cdot \frac{- 191}{63}$

Step 11.2.1.16

Multiply $\frac{8}{243 e^{\frac{191}{72}}}$ by $\frac{- 191}{63}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{8 \cdot - 191}{243 e^{\frac{191}{72}} \cdot 63}$

Step 11.2.1.17

Multiply $8$ by $- 191$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{- 1528}{243 e^{\frac{191}{72}} \cdot 63}$

Step 11.2.1.18

Multiply $63$ by $243$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} + \frac{- 1528}{15309 e^{\frac{191}{72}}}$

Step 11.2.1.19

Move the negative in front of the fraction.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} - \frac{1528}{15309 e^{\frac{191}{72}}}$

Step 11.2.2

To write $\frac{136}{2187 e^{\frac{191}{72}}}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136}{2187 e^{\frac{191}{72}}} \cdot \frac{7}{7} - \frac{1528}{15309 e^{\frac{191}{72}}}$

Step 11.2.3

Write each expression with a common denominator of $15309 e^{\frac{191}{72}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.2.3.1

Multiply $\frac{136}{2187 e^{\frac{191}{72}}}$ by $\frac{7}{7}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136 \cdot 7}{2187 e^{\frac{191}{72}} \cdot 7} - \frac{1528}{15309 e^{\frac{191}{72}}}$

Step 11.2.3.2

Multiply $7$ by $2187$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136 \cdot 7}{15309 e^{\frac{191}{72}}} - \frac{1528}{15309 e^{\frac{191}{72}}}$

Step 11.2.4

Combine the numerators over the common denominator.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{136 \cdot 7 - 1528}{15309 e^{\frac{191}{72}}}$

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Multiply $136$ by $7$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{952 - 1528}{15309 e^{\frac{191}{72}}}$

Step 11.2.5.2

Subtract $1528$ from $952$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{- 576}{15309 e^{\frac{191}{72}}}$

Step 11.2.6

Factor $9$ out of $- 576$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{9 (- 64)}{15309 e^{\frac{191}{72}}}$

Step 11.2.7

Cancel the common factors.

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Step 11.2.7.1

Factor $9$ out of $15309 e^{\frac{191}{72}}$ .

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{9 (- 64)}{9 (1701 e^{\frac{191}{72}})}$

Step 11.2.7.2

Cancel the common factor.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{9 \cdot - 64}{9 (1701 e^{\frac{191}{72}})}$

Step 11.2.7.3

Rewrite the expression.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = \frac{- 64}{1701 e^{\frac{191}{72}}}$

Step 11.2.8

Move the negative in front of the fraction.

$f (\frac{1}{3 e^{\frac{191}{504}}}) = - \frac{64}{1701 e^{\frac{191}{72}}}$

Step 11.2.9

The final answer is $- \frac{64}{1701 e^{\frac{191}{72}}}$ .

$y = - \frac{64}{1701 e^{\frac{191}{72}}}$

Step 12

These are the local extrema for $f (x) = 136 x^{7} + 576 x^{7} \ln (3 x)$ .

$(\frac{1}{3 e^{\frac{191}{504}}}, - \frac{64}{1701 e^{\frac{191}{72}}})$ is a local minima

Step 13