Calculus Examples

$f (x) = x^{2} + \frac{54}{x} - 2$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{2} + \frac{54}{x} - 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [- 2]$

Step 1.1.1.2

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

$2 x + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [- 2]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{54}{x}]$ .

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Step 1.1.2.1

Since $54$ is constant with respect to $x$ , the derivative of $\frac{54}{x}$ with respect to $x$ is $54 \frac{d}{d x} [\frac{1}{x}]$ .

$2 x + 54 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [- 2]$

Step 1.1.2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$2 x + 54 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [- 2]$

Step 1.1.2.3

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

$2 x + 54 (- x^{- 2}) + \frac{d}{d x} [- 2]$

Step 1.1.2.4

Multiply $- 1$ by $54$ .

$2 x - 54 x^{- 2} + \frac{d}{d x} [- 2]$

Step 1.1.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$2 x - 54 x^{- 2} + 0$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 x - 54 \frac{1}{x^{2}} + 0$

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

Combine $- 54$ and $\frac{1}{x^{2}}$ .

$2 x + \frac{- 54}{x^{2}} + 0$

Step 1.1.4.2.2

Move the negative in front of the fraction.

$2 x - \frac{54}{x^{2}} + 0$

Step 1.1.4.2.3

Add $2 x - \frac{54}{x^{2}}$ and $0$ .

$f' (x) = 2 x - \frac{54}{x^{2}}$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $2 x - \frac{54}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{54}{x^{2}}]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{54}{x^{2}}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.2.2.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{54}{x^{2}}]$

Step 1.2.2.2

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

$2 \cdot 1 + \frac{d}{d x} [- \frac{54}{x^{2}}]$

Step 1.2.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- \frac{54}{x^{2}}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- \frac{54}{x^{2}}]$ .

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Step 1.2.3.1

Since $- 54$ is constant with respect to $x$ , the derivative of $- \frac{54}{x^{2}}$ with respect to $x$ is $- 54 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$2 - 54 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.2.3.2

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

$2 - 54 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 1.2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 1.2.3.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$2 - 54 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

Step 1.2.3.3.2

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

$2 - 54 (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.2.3.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$2 - 54 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.2.3.4

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

$2 - 54 (- {(x^{2})}^{- 2} (2 x))$

Step 1.2.3.5

Multiply the exponents in ${(x^{2})}^{- 2}$ .

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Step 1.2.3.5.1

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

$2 - 54 (- x^{2 \cdot - 2} (2 x))$

Step 1.2.3.5.2

Multiply $2$ by $- 2$ .

$2 - 54 (- x^{- 4} (2 x))$

Step 1.2.3.6

Multiply $2$ by $- 1$ .

$2 - 54 (- 2 x^{- 4} x)$

Step 1.2.3.7

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

$2 - 54 (- 2 (x^{1} x^{- 4}))$

Step 1.2.3.8

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

$2 - 54 (- 2 x^{1 - 4})$

Step 1.2.3.9

Subtract $4$ from $1$ .

$2 - 54 (- 2 x^{- 3})$

Step 1.2.3.10

Multiply $- 2$ by $- 54$ .

$2 + 108 x^{- 3}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 + 108 \frac{1}{x^{3}}$

Step 1.2.4.2

Combine $108$ and $\frac{1}{x^{3}}$ .

$2 + \frac{108}{x^{3}}$

Step 1.2.4.3

Reorder terms.

$f'' (x) = \frac{108}{x^{3}} + 2$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{108}{x^{3}} + 2$ .

$\frac{108}{x^{3}} + 2$

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{108}{x^{3}} + 2 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{108}{x^{3}} + 2 = 0$

Step 2.2

Subtract $2$ from both sides of the equation.

$\frac{108}{x^{3}} = - 2$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{3}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 2.4

Multiply each term in $\frac{108}{x^{3}} = - 2$ by $x^{3}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{108}{x^{3}} = - 2$ by $x^{3}$ .

$\frac{108}{x^{3}} x^{3} = - 2 x^{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $x^{3}$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{108}{x^{3}} x^{3} = - 2 x^{3}$

Step 2.4.2.1.2

Rewrite the expression.

$108 = - 2 x^{3}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- 2 x^{3} = 108$ .

$- 2 x^{3} = 108$

Step 2.5.2

Subtract $108$ from both sides of the equation.

$- 2 x^{3} - 108 = 0$

Step 2.5.3

Factor $- 2$ out of $- 2 x^{3} - 108$ .

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Step 2.5.3.1

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

$- 2 x^{3} - 108 = 0$

Step 2.5.3.2

Factor $- 2$ out of $- 108$ .

$- 2 x^{3} - 2 \cdot 54 = 0$

Step 2.5.3.3

Factor $- 2$ out of $- 2 (x^{3}) - 2 (54)$ .

$- 2 (x^{3} + 54) = 0$

Step 2.5.4

Divide each term in $- 2 (x^{3} + 54) = 0$ by $- 2$ and simplify.

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Step 2.5.4.1

Divide each term in $- 2 (x^{3} + 54) = 0$ by $- 2$ .

$\frac{- 2 (x^{3} + 54)}{- 2} = \frac{0}{- 2}$

Step 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $- 2$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

$\frac{- 2 (x^{3} + 54)}{- 2} = \frac{0}{- 2}$

Step 2.5.4.2.1.2

Divide $x^{3} + 54$ by $1$ .

$x^{3} + 54 = \frac{0}{- 2}$

Step 2.5.4.3

Simplify the right side.

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Step 2.5.4.3.1

Divide $0$ by $- 2$ .

$x^{3} + 54 = 0$

Step 2.5.5

Subtract $54$ from both sides of the equation.

$x^{3} = - 54$

Step 2.5.6

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{- 54}$

Step 2.5.7

Simplify $\sqrt[3]{- 54}$ .

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Step 2.5.7.1

Rewrite $- 54$ as ${(- 3)}^{3} \cdot 2$ .

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Step 2.5.7.1.1

Factor $- 27$ out of $- 54$ .

$x = \sqrt[3]{- 27 (2)}$

Step 2.5.7.1.2

Rewrite $- 27$ as ${(- 3)}^{3}$ .

$x = \sqrt[3]{{(- 3)}^{3} \cdot 2}$

Step 2.5.7.2

Pull terms out from under the radical.

$x = - 3 \sqrt[3]{2}$

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $- 3 \sqrt[3]{2}$ in $f (x) = x^{2} + \frac{54}{x} - 2$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $- 3 \sqrt[3]{2}$ in the expression.

$f (- 3 \sqrt[3]{2}) = {(- 3 \sqrt[3]{2})}^{2} + \frac{54}{- 3 \sqrt[3]{2}} - 2$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Apply the product rule to $- 3 \sqrt[3]{2}$ .

$f (- 3 \sqrt[3]{2}) = {(- 3)}^{2} {\sqrt[3]{2}}^{2} + \frac{54}{- 3 \sqrt[3]{2}} - 2$

Step 3.1.2.1.2

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

$f (- 3 \sqrt[3]{2}) = 9 {\sqrt[3]{2}}^{2} + \frac{54}{- 3 \sqrt[3]{2}} - 2$

Step 3.1.2.1.3

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{2^{2}} + \frac{54}{- 3 \sqrt[3]{2}} - 2$

Step 3.1.2.1.4

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

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} + \frac{54}{- 3 \sqrt[3]{2}} - 2$

Step 3.1.2.1.5

Cancel the common factor of $54$ and $- 3$ .

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Step 3.1.2.1.5.1

Factor $3$ out of $54$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} + \frac{3 (18)}{- 3 \sqrt[3]{2}} - 2$

Step 3.1.2.1.5.2

Cancel the common factors.

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Step 3.1.2.1.5.2.1

Factor $3$ out of $- 3 \sqrt[3]{2}$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} + \frac{3 (18)}{3 (- \sqrt[3]{2})} - 2$

Step 3.1.2.1.5.2.2

Cancel the common factor.

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} + \frac{3 \cdot 18}{3 (- \sqrt[3]{2})} - 2$

Step 3.1.2.1.5.2.3

Rewrite the expression.

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} + \frac{18}{- \sqrt[3]{2}} - 2$

Step 3.1.2.1.6

Move the negative in front of the fraction.

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{18}{\sqrt[3]{2}} - 2$

Step 3.1.2.1.7

Multiply $\frac{18}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - (\frac{18}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}) - 2$

Step 3.1.2.1.8

Combine and simplify the denominator.

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Step 3.1.2.1.8.1

Multiply $\frac{18}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{18 {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}} - 2$

Step 3.1.2.1.8.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{18 {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}} - 2$

Step 3.1.2.1.8.3

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

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{18 {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}} - 2$

Step 3.1.2.1.8.4

Add $1$ and $2$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{18 {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}} - 2$

Step 3.1.2.1.8.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 3.1.2.1.8.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{18 {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}} - 2$

Step 3.1.2.1.8.5.2

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

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{18 {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}} - 2$

Step 3.1.2.1.8.5.3

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

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{18 {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}} - 2$

Step 3.1.2.1.8.5.4

Cancel the common factor of $3$ .

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Step 3.1.2.1.8.5.4.1

Cancel the common factor.

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{18 {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}} - 2$

Step 3.1.2.1.8.5.4.2

Rewrite the expression.

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{18 {\sqrt[3]{2}}^{2}}{2} - 2$

Step 3.1.2.1.8.5.5

Evaluate the exponent.

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{18 {\sqrt[3]{2}}^{2}}{2} - 2$

Step 3.1.2.1.9

Cancel the common factor of $18$ and $2$ .

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Step 3.1.2.1.9.1

Factor $2$ out of $18 {\sqrt[3]{2}}^{2}$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{2 (9 {\sqrt[3]{2}}^{2})}{2} - 2$

Step 3.1.2.1.9.2

Cancel the common factors.

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Step 3.1.2.1.9.2.1

Factor $2$ out of $2$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{2 (9 {\sqrt[3]{2}}^{2})}{2 (1)} - 2$

Step 3.1.2.1.9.2.2

Cancel the common factor.

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{2 (9 {\sqrt[3]{2}}^{2})}{2 \cdot 1} - 2$

Step 3.1.2.1.9.2.3

Rewrite the expression.

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - \frac{9 {\sqrt[3]{2}}^{2}}{1} - 2$

Step 3.1.2.1.9.2.4

Divide $9 {\sqrt[3]{2}}^{2}$ by $1$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - (9 {\sqrt[3]{2}}^{2}) - 2$

Step 3.1.2.1.10

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - (9 \sqrt[3]{2^{2}}) - 2$

Step 3.1.2.1.11

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

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - (9 \sqrt[3]{4}) - 2$

Step 3.1.2.1.12

Multiply $9$ by $- 1$ .

$f (- 3 \sqrt[3]{2}) = 9 \sqrt[3]{4} - 9 \sqrt[3]{4} - 2$

Step 3.1.2.2

Simplify by adding terms.

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Step 3.1.2.2.1

Subtract $9 \sqrt[3]{4}$ from $9 \sqrt[3]{4}$ .

$f (- 3 \sqrt[3]{2}) = 0 - 2$

Step 3.1.2.2.2

Subtract $2$ from $0$ .

$f (- 3 \sqrt[3]{2}) = - 2$

Step 3.1.2.3

The final answer is $- 2$ .

$- 2$

Step 3.2

The point found by substituting $- 3 \sqrt[3]{2}$ in $f (x) = x^{2} + \frac{54}{x} - 2$ is $(- 3 \sqrt[3]{2}, - 2)$ . This point can be an inflection point.

$(- 3 \sqrt[3]{2}, - 2)$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - 3 \sqrt[3]{2}) \cup (- 3 \sqrt[3]{2}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 3 \sqrt[3]{2})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 3.87976314$ in the expression.

$f'' (- 3.87976314) = \frac{108}{{(- 3.87976314)}^{3}} + 2$

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

Raise $- 3.87976314$ to the power of $3$ .

$f'' (- 3.87976314) = \frac{108}{- 58.40037573} + 2$

Step 5.2.1.2

Divide $108$ by $- 58.40037573$ .

$f'' (- 3.87976314) = - 1.84930317 + 2$

Step 5.2.2

Add $- 1.84930317$ and $2$ .

$f'' (- 3.87976314) = 0.15069682$

Step 5.2.3

The final answer is $0.15069682$ .

$0.15069682$

Step 5.3

At $- 3.87976314$ , the second derivative is $0.15069682$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 3 \sqrt[3]{2})$ .

Increasing on $(- \infty, - 3 \sqrt[3]{2})$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- 3 \sqrt[3]{2}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 3.67976314$ in the expression.

$f'' (- 3.67976314) = \frac{108}{{(- 3.67976314)}^{3}} + 2$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 3.67976314$ to the power of $3$ .

$f'' (- 3.67976314) = \frac{108}{- 49.82641005} + 2$

Step 6.2.1.2

Divide $108$ by $- 49.82641005$ .

$f'' (- 3.67976314) = - 2.16752521 + 2$

Step 6.2.2

Add $- 2.16752521$ and $2$ .

$f'' (- 3.67976314) = - 0.16752521$

Step 6.2.3

The final answer is $- 0.16752521$ .

$- 0.16752521$

Step 6.3

At $- 3.67976314$ , the second derivative is $- 0.16752521$ . Since this is negative, the second derivative is decreasing on the interval $(- 3 \sqrt[3]{2}, \infty)$

Decreasing on $(- 3 \sqrt[3]{2}, \infty)$ since $f'' (x) < 0$

Step 7

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(- 3 \sqrt[3]{2}, - 2)$ .

$(- 3 \sqrt[3]{2}, - 2)$

Step 8