Calculus Examples

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

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

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 1.1.2

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

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Step 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 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 1.1.2.3

Multiply $2$ by $64$ .

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

Step 1.1.3

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

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Step 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 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 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 1.1.3.4

Multiply $- 1$ by $54$ .

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

Step 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 1.1.5

Simplify.

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Step 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 1.1.5.2

Combine terms.

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Step 1.1.5.2.1

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

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

Step 1.1.5.2.2

Move the negative in front of the fraction.

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

Step 1.1.5.2.3

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

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

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $128 x - \frac{54}{x^{2}}$ 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 1.2.2

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

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Step 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 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 1.2.2.3

Multiply $128$ by $1$ .

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

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

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

Step 1.2.3.2

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

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

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 $x^{2}$ .

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

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 $n = - 1$ .

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

Step 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 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 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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.3.5.2

Multiply $2$ by $- 2$ .

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

Step 1.2.3.6

Multiply $2$ by $- 1$ .

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

Step 1.2.3.7

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

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

Step 1.2.3.8

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

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

Step 1.2.3.9

Subtract $4$ from $1$ .

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

Step 1.2.3.10

Multiply $- 2$ by $- 54$ .

$128 + 108 x^{- 3}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.4.3

Reorder terms.

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

Step 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

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

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Step 2.1

Set the second derivative equal to $0$ .

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

Step 2.2

Subtract $128$ from both sides of the equation.

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

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

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Step 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 2.4.2

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.

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

Step 2.4.2.1.2

Rewrite the expression.

$108 = - 128 x^{3}$

Step 2.5

Solve the equation.

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Step 2.5.1

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

$- 128 x^{3} = 108$

Step 2.5.2

Subtract $108$ from both sides of the equation.

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

Step 2.5.3

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

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Step 2.5.3.1

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

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

Step 2.5.3.2

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

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

Step 2.5.3.3

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

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

Step 2.5.4

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

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Step 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 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $- 4$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

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

Step 2.5.4.2.1.2

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

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

Step 2.5.4.3

Simplify the right side.

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Step 2.5.4.3.1

Divide $0$ by $- 4$ .

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

Step 2.5.5

Subtract $27$ from both sides of the equation.

$32 x^{3} = - 27$

Step 2.5.6

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

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Step 2.5.6.1

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

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

Step 2.5.6.2

Simplify the left side.

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Step 2.5.6.2.1

Cancel the common factor of $32$ .

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Step 2.5.6.2.1.1

Cancel the common factor.

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

Step 2.5.6.2.1.2

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

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

Step 2.5.6.3

Simplify the right side.

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Step 2.5.6.3.1

Move the negative in front of the fraction.

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

Step 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 2.5.8

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

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Step 2.5.8.1

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

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Step 2.5.8.1.1

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

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

Step 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 2.5.8.1.3

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

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

Step 2.5.8.1.4

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 2.5.8.1.5

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

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

Step 2.5.8.2

Pull terms out from under the radical.

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

Step 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 2.5.8.4

Any root of $1$ is $1$ .

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

Step 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 2.5.8.6

Combine and simplify the denominator.

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Step 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 2.5.8.6.2

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 2.5.8.6.3

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 2.5.8.6.4

Add $1$ and $2$ .

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

Step 2.5.8.6.5

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

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Step 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 2.5.8.6.5.2

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 2.5.8.6.5.3

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

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

Step 2.5.8.6.5.4

Cancel the common factor of $3$ .

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Step 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 2.5.8.6.5.4.2

Rewrite the expression.

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

Step 2.5.8.6.5.5

Evaluate the exponent.

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

Step 2.5.8.7

Simplify the numerator.

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Step 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 2.5.8.7.2

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

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

Step 2.5.8.7.3

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

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Step 2.5.8.7.3.1

Factor $8$ out of $16$ .

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

Step 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 2.5.8.7.4

Pull terms out from under the radical.

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

Step 2.5.8.8

Simplify terms.

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Step 2.5.8.8.1

Cancel the common factor of $2$ .

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Step 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 2.5.8.8.1.2

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

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

Step 2.5.8.8.1.3

Cancel the common factor.

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

Step 2.5.8.8.1.4

Rewrite the expression.

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

Step 2.5.8.8.2

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

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

Step 2.5.8.8.3

Move the negative in front of the fraction.

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

Step 3

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

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Step 3.1

Substitute $- \frac{3 \sqrt[3]{2}}{4}$ in $f (x) = 64 x^{2} + \frac{54}{x} + 2$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $- \frac{3 \sqrt[3]{2}}{4}$ in the expression.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 64 {(- \frac{3 \sqrt[3]{2}}{4})}^{2} + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

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

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

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Step 3.1.2.1.1.1

Apply the product rule to $- \frac{3 \sqrt[3]{2}}{4}$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 64 ({(- 1)}^{2} {(\frac{3 \sqrt[3]{2}}{4})}^{2}) + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.1.2

Apply the product rule to $\frac{3 \sqrt[3]{2}}{4}$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 64 ({(- 1)}^{2} (\frac{{(3 \sqrt[3]{2})}^{2}}{4^{2}})) + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.1.3

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 64 ({(- 1)}^{2} (\frac{3^{2} {\sqrt[3]{2}}^{2}}{4^{2}})) + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.2

Raise $- 1$ to the power of $2$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 64 (1 (\frac{3^{2} {\sqrt[3]{2}}^{2}}{4^{2}})) + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.3

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 64 (\frac{3^{2} {\sqrt[3]{2}}^{2}}{4^{2}}) + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.4

Simplify the numerator.

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Step 3.1.2.1.4.1

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 64 (\frac{9 {\sqrt[3]{2}}^{2}}{4^{2}}) + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.4.2

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 64 (\frac{9 \sqrt[3]{2^{2}}}{4^{2}}) + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.4.3

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 64 (\frac{9 \sqrt[3]{4}}{4^{2}}) + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.5

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 64 (\frac{9 \sqrt[3]{4}}{16}) + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.6

Cancel the common factor of $16$ .

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Step 3.1.2.1.6.1

Factor $16$ out of $64$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 16 (4) (\frac{9 \sqrt[3]{4}}{16}) + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.6.2

Cancel the common factor.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 16 \cdot (4 (\frac{9 \sqrt[3]{4}}{16})) + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.6.3

Rewrite the expression.

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

Step 3.1.2.1.7

Multiply $9$ by $4$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} + \frac{54}{- \frac{3 \sqrt[3]{2}}{4}} + 2$

Step 3.1.2.1.8

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} + 54 (- \frac{4}{3 \sqrt[3]{2}}) + 2$

Step 3.1.2.1.9

Cancel the common factor of $3$ .

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Step 3.1.2.1.9.1

Move the leading negative in $- \frac{4}{3 \sqrt[3]{2}}$ into the numerator.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} + 54 (\frac{- 4}{3 \sqrt[3]{2}}) + 2$

Step 3.1.2.1.9.2

Factor $3$ out of $54$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} + 3 (18) (\frac{- 4}{3 \sqrt[3]{2}}) + 2$

Step 3.1.2.1.9.3

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} + 3 (18) (\frac{- 4}{3 (\sqrt[3]{2})}) + 2$

Step 3.1.2.1.9.4

Cancel the common factor.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} + 3 \cdot (18 (\frac{- 4}{3 \sqrt[3]{2}})) + 2$

Step 3.1.2.1.9.5

Rewrite the expression.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} + 18 (\frac{- 4}{\sqrt[3]{2}}) + 2$

Step 3.1.2.1.10

Combine $18$ and $\frac{- 4}{\sqrt[3]{2}}$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} + \frac{18 \cdot - 4}{\sqrt[3]{2}} + 2$

Step 3.1.2.1.11

Multiply $18$ by $- 4$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} + \frac{- 72}{\sqrt[3]{2}} + 2$

Step 3.1.2.1.12

Move the negative in front of the fraction.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{72}{\sqrt[3]{2}} + 2$

Step 3.1.2.1.13

Multiply $\frac{72}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - (\frac{72}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}) + 2$

Step 3.1.2.1.14

Combine and simplify the denominator.

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Step 3.1.2.1.14.1

Multiply $\frac{72}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{72 {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}} + 2$

Step 3.1.2.1.14.2

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{72 {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}} + 2$

Step 3.1.2.1.14.3

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{72 {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}} + 2$

Step 3.1.2.1.14.4

Add $1$ and $2$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{72 {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}} + 2$

Step 3.1.2.1.14.5

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

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Step 3.1.2.1.14.5.1

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{72 {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}} + 2$

Step 3.1.2.1.14.5.2

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{72 {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}} + 2$

Step 3.1.2.1.14.5.3

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{72 {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}} + 2$

Step 3.1.2.1.14.5.4

Cancel the common factor of $3$ .

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Step 3.1.2.1.14.5.4.1

Cancel the common factor.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{72 {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}} + 2$

Step 3.1.2.1.14.5.4.2

Rewrite the expression.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{72 {\sqrt[3]{2}}^{2}}{2} + 2$

Step 3.1.2.1.14.5.5

Evaluate the exponent.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{72 {\sqrt[3]{2}}^{2}}{2} + 2$

Step 3.1.2.1.15

Cancel the common factor of $72$ and $2$ .

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Step 3.1.2.1.15.1

Factor $2$ out of $72 {\sqrt[3]{2}}^{2}$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{2 (36 {\sqrt[3]{2}}^{2})}{2} + 2$

Step 3.1.2.1.15.2

Cancel the common factors.

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Step 3.1.2.1.15.2.1

Factor $2$ out of $2$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{2 (36 {\sqrt[3]{2}}^{2})}{2 (1)} + 2$

Step 3.1.2.1.15.2.2

Cancel the common factor.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{2 (36 {\sqrt[3]{2}}^{2})}{2 \cdot 1} + 2$

Step 3.1.2.1.15.2.3

Rewrite the expression.

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - \frac{36 {\sqrt[3]{2}}^{2}}{1} + 2$

Step 3.1.2.1.15.2.4

Divide $36 {\sqrt[3]{2}}^{2}$ by $1$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - (36 {\sqrt[3]{2}}^{2}) + 2$

Step 3.1.2.1.16

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - (36 \sqrt[3]{2^{2}}) + 2$

Step 3.1.2.1.17

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

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - (36 \sqrt[3]{4}) + 2$

Step 3.1.2.1.18

Multiply $36$ by $- 1$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 36 \sqrt[3]{4} - 36 \sqrt[3]{4} + 2$

Step 3.1.2.2

Simplify by adding terms.

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Step 3.1.2.2.1

Subtract $36 \sqrt[3]{4}$ from $36 \sqrt[3]{4}$ .

$f (- \frac{3 \sqrt[3]{2}}{4}) = 0 + 2$

Step 3.1.2.2.2

Add $0$ and $2$ .

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

Step 3.1.2.3

The final answer is $2$ .

$2$

Step 3.2

The point found by substituting $- \frac{3 \sqrt[3]{2}}{4}$ in $f (x) = 64 x^{2} + \frac{54}{x} + 2$ is $(- \frac{3 \sqrt[3]{2}}{4}, 2)$ . This point can be an inflection point.

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

Step 4

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

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

Step 5

Substitute a value from the interval $(- \infty, - \frac{3 \sqrt[3]{2}}{4})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1.04494078$ in the expression.

$f'' (- 1.04494078) = \frac{108}{{(- 1.04494078)}^{3}} + 128$

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 $- 1.04494078$ to the power of $3$ .

$f'' (- 1.04494078) = \frac{108}{- 1.14097215} + 128$

Step 5.2.1.2

Divide $108$ by $- 1.14097215$ .

$f'' (- 1.04494078) = - 94.65612275 + 128$

Step 5.2.2

Add $- 94.65612275$ and $128$ .

$f'' (- 1.04494078) = 33.34387724$

Step 5.2.3

The final answer is $33.34387724$ .

$33.34387724$

Step 5.3

At $- 1.04494078$ , the second derivative is $33.34387724$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - \frac{3 \sqrt[3]{2}}{4})$ .

Increasing on $(- \infty, - \frac{3 \sqrt[3]{2}}{4})$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- \frac{3 \sqrt[3]{2}}{4}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.84494078$ in the expression.

$f'' (- 0.84494078) = \frac{108}{{(- 0.84494078)}^{3}} + 128$

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 $- 0.84494078$ to the power of $3$ .

$f'' (- 0.84494078) = \frac{108}{- 0.60322429} + 128$

Step 6.2.1.2

Divide $108$ by $- 0.60322429$ .

$f'' (- 0.84494078) = - 179.03788142 + 128$

Step 6.2.2

Add $- 179.03788142$ and $128$ .

$f'' (- 0.84494078) = - 51.03788142$

Step 6.2.3

The final answer is $- 51.03788142$ .

$- 51.03788142$

Step 6.3

At $- 0.84494078$ , the second derivative is $- 51.03788142$ . Since this is negative, the second derivative is decreasing on the interval $(- \frac{3 \sqrt[3]{2}}{4}, \infty)$

Decreasing on $(- \frac{3 \sqrt[3]{2}}{4}, \infty)$ 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 $(- \frac{3 \sqrt[3]{2}}{4}, 2)$ .

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

Step 8