Calculus Examples

$64 x^{2} + \frac{54}{x} + 2$

Step 1

Write $64 x^{2} + \frac{54}{x} + 2$ as a function.

$f (x) = 64 x^{2} + \frac{54}{x} + 2$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

By the Sum Rule, the derivative of $64 x^{2} + \frac{54}{x} + 2$ with respect to $x$ is $\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [2]$

Step 2.1.1.2

Evaluate $\frac{d}{d x} [64 x^{2}]$ .

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Step 2.1.1.2.1

Since $64$ is constant with respect to $x$ , the derivative of $64 x^{2}$ with respect to $x$ is $64 \frac{d}{d x} [x^{2}]$ .

$64 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [2]$

Step 2.1.1.2.2

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

$64 (2 x) + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [2]$

Step 2.1.1.2.3

Multiply $2$ by $64$ .

$128 x + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [2]$

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

$128 x + 54 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [2]$

Step 2.1.1.3.2

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

$128 x + 54 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [2]$

Step 2.1.1.3.3

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

$128 x + 54 (- x^{- 2}) + \frac{d}{d x} [2]$

Step 2.1.1.3.4

Multiply $- 1$ by $54$ .

$128 x - 54 x^{- 2} + \frac{d}{d x} [2]$

Step 2.1.1.4

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

$128 x - 54 x^{- 2} + 0$

Step 2.1.1.5

Simplify.

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Step 2.1.1.5.1

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

$128 x - 54 \frac{1}{x^{2}} + 0$

Step 2.1.1.5.2

Combine terms.

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Step 2.1.1.5.2.1

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

$128 x + \frac{- 54}{x^{2}} + 0$

Step 2.1.1.5.2.2

Move the negative in front of the fraction.

$128 x - \frac{54}{x^{2}} + 0$

Step 2.1.1.5.2.3

Add $128 x - \frac{54}{x^{2}}$ and $0$ .

$f' (x) = 128 x - \frac{54}{x^{2}}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $128 x - \frac{54}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [128 x] + \frac{d}{d x} [- \frac{54}{x^{2}}]$ .

$\frac{d}{d x} [128 x] + \frac{d}{d x} [- \frac{54}{x^{2}}]$

Step 2.1.2.2

Evaluate $\frac{d}{d x} [128 x]$ .

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Step 2.1.2.2.1

Since $128$ is constant with respect to $x$ , the derivative of $128 x$ with respect to $x$ is $128 \frac{d}{d x} [x]$ .

$128 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{54}{x^{2}}]$

Step 2.1.2.2.2

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

$128 \cdot 1 + \frac{d}{d x} [- \frac{54}{x^{2}}]$

Step 2.1.2.2.3

Multiply $128$ by $1$ .

$128 + \frac{d}{d x} [- \frac{54}{x^{2}}]$

Step 2.1.2.3

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

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Step 2.1.2.3.1

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

$128 - 54 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2.1.2.3.2

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

$128 - 54 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 2.1.2.3.3

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 2.1.2.3.3.1

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

$128 - 54 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

Step 2.1.2.3.3.2

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

$128 - 54 (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 2.1.2.3.3.3

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

$128 - 54 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 2.1.2.3.4

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

$128 - 54 (- {(x^{2})}^{- 2} (2 x))$

Step 2.1.2.3.5

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

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Step 2.1.2.3.5.1

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

$128 - 54 (- x^{2 \cdot - 2} (2 x))$

Step 2.1.2.3.5.2

Multiply $2$ by $- 2$ .

$128 - 54 (- x^{- 4} (2 x))$

Step 2.1.2.3.6

Multiply $2$ by $- 1$ .

$128 - 54 (- 2 x^{- 4} x)$

Step 2.1.2.3.7

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

$128 - 54 (- 2 (x^{1} x^{- 4}))$

Step 2.1.2.3.8

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

$128 - 54 (- 2 x^{1 - 4})$

Step 2.1.2.3.9

Subtract $4$ from $1$ .

$128 - 54 (- 2 x^{- 3})$

Step 2.1.2.3.10

Multiply $- 2$ by $- 54$ .

$128 + 108 x^{- 3}$

Step 2.1.2.4

Simplify.

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Step 2.1.2.4.1

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

$128 + 108 \frac{1}{x^{3}}$

Step 2.1.2.4.2

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

$128 + \frac{108}{x^{3}}$

Step 2.1.2.4.3

Reorder terms.

$f'' (x) = \frac{108}{x^{3}} + 128$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{108}{x^{3}} + 128$ .

$\frac{108}{x^{3}} + 128$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{108}{x^{3}} + 128 = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{108}{x^{3}} + 128 = 0$

Step 2.2.2

Subtract $128$ from both sides of the equation.

$\frac{108}{x^{3}} = - 128$

Step 2.2.3

Find the LCD of the terms in the equation.

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Step 2.2.3.1

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 2.2.3.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 2.2.4

Multiply each term in $\frac{108}{x^{3}} = - 128$ by $x^{3}$ to eliminate the fractions.

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Step 2.2.4.1

Multiply each term in $\frac{108}{x^{3}} = - 128$ by $x^{3}$ .

$\frac{108}{x^{3}} x^{3} = - 128 x^{3}$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

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

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Step 2.2.4.2.1.1

Cancel the common factor.

$\frac{108}{x^{3}} x^{3} = - 128 x^{3}$

Step 2.2.4.2.1.2

Rewrite the expression.

$108 = - 128 x^{3}$

Step 2.2.5

Solve the equation.

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Step 2.2.5.1

Rewrite the equation as $- 128 x^{3} = 108$ .

$- 128 x^{3} = 108$

Step 2.2.5.2

Subtract $108$ from both sides of the equation.

$- 128 x^{3} - 108 = 0$

Step 2.2.5.3

Factor $- 4$ out of $- 128 x^{3} - 108$ .

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Step 2.2.5.3.1

Factor $- 4$ out of $- 128 x^{3}$ .

$- 4 (32 x^{3}) - 108 = 0$

Step 2.2.5.3.2

Factor $- 4$ out of $- 108$ .

$- 4 (32 x^{3}) - 4 \cdot 27 = 0$

Step 2.2.5.3.3

Factor $- 4$ out of $- 4 (32 x^{3}) - 4 (27)$ .

$- 4 (32 x^{3} + 27) = 0$

Step 2.2.5.4

Divide each term in $- 4 (32 x^{3} + 27) = 0$ by $- 4$ and simplify.

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Step 2.2.5.4.1

Divide each term in $- 4 (32 x^{3} + 27) = 0$ by $- 4$ .

$\frac{- 4 (32 x^{3} + 27)}{- 4} = \frac{0}{- 4}$

Step 2.2.5.4.2

Simplify the left side.

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Step 2.2.5.4.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.5.4.2.1.1

Cancel the common factor.

$\frac{- 4 (32 x^{3} + 27)}{- 4} = \frac{0}{- 4}$

Step 2.2.5.4.2.1.2

Divide $32 x^{3} + 27$ by $1$ .

$32 x^{3} + 27 = \frac{0}{- 4}$

Step 2.2.5.4.3

Simplify the right side.

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Step 2.2.5.4.3.1

Divide $0$ by $- 4$ .

$32 x^{3} + 27 = 0$

Step 2.2.5.5

Subtract $27$ from both sides of the equation.

$32 x^{3} = - 27$

Step 2.2.5.6

Divide each term in $32 x^{3} = - 27$ by $32$ and simplify.

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Step 2.2.5.6.1

Divide each term in $32 x^{3} = - 27$ by $32$ .

$\frac{32 x^{3}}{32} = \frac{- 27}{32}$

Step 2.2.5.6.2

Simplify the left side.

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Step 2.2.5.6.2.1

Cancel the common factor of $32$ .

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Step 2.2.5.6.2.1.1

Cancel the common factor.

$\frac{32 x^{3}}{32} = \frac{- 27}{32}$

Step 2.2.5.6.2.1.2

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

$x^{3} = \frac{- 27}{32}$

Step 2.2.5.6.3

Simplify the right side.

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Step 2.2.5.6.3.1

Move the negative in front of the fraction.

$x^{3} = - \frac{27}{32}$

Step 2.2.5.7

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

$x = \sqrt[3]{- \frac{27}{32}}$

Step 2.2.5.8

Simplify $\sqrt[3]{- \frac{27}{32}}$ .

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Step 2.2.5.8.1

Rewrite $- \frac{27}{32}$ as ${(- \frac{3}{2})}^{3} \frac{1}{4}$ .

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Step 2.2.5.8.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \frac{27}{32}}$

Step 2.2.5.8.1.2

Factor the perfect power $3^{3}$ out of $27$ .

$x = \sqrt[3]{{(- 1)}^{3} \frac{3^{3} \cdot 1}{32}}$

Step 2.2.5.8.1.3

Factor the perfect power $2^{3}$ out of $32$ .

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

Step 2.2.5.8.1.4

Rearrange the fraction $\frac{3^{3} \cdot 1}{2^{3} \cdot 4}$ .

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

Step 2.2.5.8.1.5

Rewrite ${(- 1)}^{3} {(\frac{3}{2})}^{3}$ as ${(- \frac{3}{2})}^{3}$ .

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

Step 2.2.5.8.2

Pull terms out from under the radical.

$x = - \frac{3}{2} \sqrt[3]{\frac{1}{4}}$

Step 2.2.5.8.3

Rewrite $\sqrt[3]{\frac{1}{4}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{4}}$ .

$x = - \frac{3}{2} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{4}}$

Step 2.2.5.8.4

Any root of $1$ is $1$ .

$x = - \frac{3}{2} \cdot \frac{1}{\sqrt[3]{4}}$

Step 2.2.5.8.5

Multiply $\frac{1}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

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

Step 2.2.5.8.6

Combine and simplify the denominator.

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Step 2.2.5.8.6.1

Multiply $\frac{1}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$x = - \frac{3}{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 2.2.5.8.6.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$x = - \frac{3}{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 2.2.5.8.6.3

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

$x = - \frac{3}{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 2.2.5.8.6.4

Add $1$ and $2$ .

$x = - \frac{3}{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 2.2.5.8.6.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 2.2.5.8.6.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

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

Step 2.2.5.8.6.5.2

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

$x = - \frac{3}{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 2.2.5.8.6.5.3

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

$x = - \frac{3}{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 2.2.5.8.6.5.4

Cancel the common factor of $3$ .

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Step 2.2.5.8.6.5.4.1

Cancel the common factor.

$x = - \frac{3}{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 2.2.5.8.6.5.4.2

Rewrite the expression.

$x = - \frac{3}{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{4^{1}}$

Step 2.2.5.8.6.5.5

Evaluate the exponent.

$x = - \frac{3}{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{4}$

Step 2.2.5.8.7

Simplify the numerator.

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Step 2.2.5.8.7.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$x = - \frac{3}{2} \cdot \frac{\sqrt[3]{4^{2}}}{4}$

Step 2.2.5.8.7.2

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

$x = - \frac{3}{2} \cdot \frac{\sqrt[3]{16}}{4}$

Step 2.2.5.8.7.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 2.2.5.8.7.3.1

Factor $8$ out of $16$ .

$x = - \frac{3}{2} \cdot \frac{\sqrt[3]{8 (2)}}{4}$

Step 2.2.5.8.7.3.2

Rewrite $8$ as $2^{3}$ .

$x = - \frac{3}{2} \cdot \frac{\sqrt[3]{2^{3} \cdot 2}}{4}$

Step 2.2.5.8.7.4

Pull terms out from under the radical.

$x = - \frac{3}{2} \cdot \frac{2 \sqrt[3]{2}}{4}$

Step 2.2.5.8.8

Simplify terms.

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Step 2.2.5.8.8.1

Cancel the common factor of $2$ .

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Step 2.2.5.8.8.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$x = \frac{- 3}{2} \cdot \frac{2 \sqrt[3]{2}}{4}$

Step 2.2.5.8.8.1.2

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

$x = \frac{- 3}{2} \cdot \frac{2 (\sqrt[3]{2})}{4}$

Step 2.2.5.8.8.1.3

Cancel the common factor.

$x = \frac{- 3}{2} \cdot \frac{2 \sqrt[3]{2}}{4}$

Step 2.2.5.8.8.1.4

Rewrite the expression.

$x = - 3 \frac{\sqrt[3]{2}}{4}$

Step 2.2.5.8.8.2

Combine $- 3$ and $\frac{\sqrt[3]{2}}{4}$ .

$x = \frac{- 3 \sqrt[3]{2}}{4}$

Step 2.2.5.8.8.3

Move the negative in front of the fraction.

$x = - \frac{3 \sqrt[3]{2}}{4}$

Step 3

Find the domain of $f (x) = 64 x^{2} + \frac{54}{x} + 2$ .

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Step 3.1

Set the denominator in $\frac{54}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 3.2

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - \frac{3 \sqrt[3]{2}}{4}) \cup (- \frac{3 \sqrt[3]{2}}{4}, 0) \cup (0, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - \frac{3 \sqrt[3]{2}}{4})$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 3$ in the expression.

$f'' (- 3) = \frac{108}{{(- 3)}^{3}} + 128$

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

$f'' (- 3) = \frac{108}{- 27} + 128$

Step 5.2.1.2

Divide $108$ by $- 27$ .

$f'' (- 3) = - 4 + 128$

Step 5.2.2

Add $- 4$ and $128$ .

$f'' (- 3) = 124$

Step 5.2.3

The final answer is $124$ .

$124$

Step 5.3

The graph is concave up on the interval $(- \infty, - \frac{3 \sqrt[3]{2}}{4})$ because $f'' (- 3)$ is positive.

Concave up on $(- \infty, - \frac{3 \sqrt[3]{2}}{4})$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(- \frac{3 \sqrt[3]{2}}{4}, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $- 0.47$ in the expression.

$f'' (- 0.47) = \frac{108}{{(- 0.47)}^{3}} + 128$

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 $- 0.47$ to the power of $3$ .

$f'' (- 0.47) = \frac{108}{- 0.103823} + 128$

Step 6.2.1.2

Divide $108$ by $- 0.103823$ .

$f'' (- 0.47) = - 1040.23193319 + 128$

Step 6.2.2

Add $- 1040.23193319$ and $128$ .

$f'' (- 0.47) = - 912.23193319$

Step 6.2.3

The final answer is $- 912.23193319$ .

$- 912.23193319$

Step 6.3

The graph is concave down on the interval $(- \frac{3 \sqrt[3]{2}}{4}, 0)$ because $f'' (- 0.47)$ is negative.

Concave down on $(- \frac{3 \sqrt[3]{2}}{4}, 0)$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(0, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $2$ in the expression.

$f'' (2) = \frac{108}{{(2)}^{3}} + 128$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

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

$f'' (2) = \frac{108}{8} + 128$

Step 7.2.1.2

Cancel the common factor of $108$ and $8$ .

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Step 7.2.1.2.1

Factor $4$ out of $108$ .

$f'' (2) = \frac{4 (27)}{8} + 128$

Step 7.2.1.2.2

Cancel the common factors.

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Step 7.2.1.2.2.1

Factor $4$ out of $8$ .

$f'' (2) = \frac{4 \cdot 27}{4 \cdot 2} + 128$

Step 7.2.1.2.2.2

Cancel the common factor.

$f'' (2) = \frac{4 \cdot 27}{4 \cdot 2} + 128$

Step 7.2.1.2.2.3

Rewrite the expression.

$f'' (2) = \frac{27}{2} + 128$

Step 7.2.2

To write $128$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (2) = \frac{27}{2} + 128 \cdot \frac{2}{2}$

Step 7.2.3

Combine $128$ and $\frac{2}{2}$ .

$f'' (2) = \frac{27}{2} + \frac{128 \cdot 2}{2}$

Step 7.2.4

Combine the numerators over the common denominator.

$f'' (2) = \frac{27 + 128 \cdot 2}{2}$

Step 7.2.5

Simplify the numerator.

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Step 7.2.5.1

Multiply $128$ by $2$ .

$f'' (2) = \frac{27 + 256}{2}$

Step 7.2.5.2

Add $27$ and $256$ .

$f'' (2) = \frac{283}{2}$

Step 7.2.6

The final answer is $\frac{283}{2}$ .

$\frac{283}{2}$

Step 7.3

The graph is concave up on the interval $(0, \infty)$ because $f'' (2)$ is positive.

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

Step 8

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, - \frac{3 \sqrt[3]{2}}{4})$ since $f'' (x)$ is positive

Concave down on $(- \frac{3 \sqrt[3]{2}}{4}, 0)$ since $f'' (x)$ is negative

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

Step 9