Calculus Examples

$f (x) = x^{2} + \frac{1}{x}$ , $x > 0$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Differentiate.

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Step 1.1.1.1.1

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

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

Step 1.1.1.1.2

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

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

Step 1.1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

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

$2 x - x^{- 2}$

Step 1.1.1.3

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

$f' (x) = 2 x - \frac{1}{x^{2}}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $2 x - \frac{1}{x^{2}}$ .

$2 x - \frac{1}{x^{2}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $2 x - \frac{1}{x^{2}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$2 x - \frac{1}{x^{2}} = 0$

Step 1.2.2

Find the LCD of the terms in the equation.

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Step 1.2.2.1

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

$1, x^{2}, 1$

Step 1.2.2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 1.2.3

Multiply each term in $2 x - \frac{1}{x^{2}} = 0$ by $x^{2}$ to eliminate the fractions.

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Step 1.2.3.1

Multiply each term in $2 x - \frac{1}{x^{2}} = 0$ by $x^{2}$ .

$2 x \cdot x^{2} - \frac{1}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify each term.

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Step 1.2.3.2.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.2.3.2.1.1.1

Move $x^{2}$ .

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

Step 1.2.3.2.1.1.2

Multiply $x^{2}$ by $x$ .

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Step 1.2.3.2.1.1.2.1

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

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

Step 1.2.3.2.1.1.2.2

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

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

Step 1.2.3.2.1.1.3

Add $2$ and $1$ .

$2 x^{3} - \frac{1}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.3.2.1.2

Cancel the common factor of $x^{2}$ .

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Step 1.2.3.2.1.2.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

$2 x^{3} + \frac{- 1}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$2 x^{3} + \frac{- 1}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$2 x^{3} - 1 = 0 x^{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Multiply $0$ by $x^{2}$ .

$2 x^{3} - 1 = 0$

Step 1.2.4

Solve the equation.

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Step 1.2.4.1

Add $1$ to both sides of the equation.

$2 x^{3} = 1$

Step 1.2.4.2

Divide each term in $2 x^{3} = 1$ by $2$ and simplify.

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Step 1.2.4.2.1

Divide each term in $2 x^{3} = 1$ by $2$ .

$\frac{2 x^{3}}{2} = \frac{1}{2}$

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1.1

Cancel the common factor.

$\frac{2 x^{3}}{2} = \frac{1}{2}$

Step 1.2.4.2.2.1.2

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

$x^{3} = \frac{1}{2}$

Step 1.2.4.3

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

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

Step 1.2.4.4

Simplify $\sqrt[3]{\frac{1}{2}}$ .

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Step 1.2.4.4.1

Rewrite $\sqrt[3]{\frac{1}{2}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{2}}$ .

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

Step 1.2.4.4.2

Any root of $1$ is $1$ .

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

Step 1.2.4.4.3

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

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

Step 1.2.4.4.4

Combine and simplify the denominator.

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Step 1.2.4.4.4.1

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

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

Step 1.2.4.4.4.2

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

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

Step 1.2.4.4.4.3

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

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

Step 1.2.4.4.4.4

Add $1$ and $2$ .

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

Step 1.2.4.4.4.5

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

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Step 1.2.4.4.4.5.1

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

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

Step 1.2.4.4.4.5.2

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

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

Step 1.2.4.4.4.5.3

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

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

Step 1.2.4.4.4.5.4

Cancel the common factor of $3$ .

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Step 1.2.4.4.4.5.4.1

Cancel the common factor.

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

Step 1.2.4.4.4.5.4.2

Rewrite the expression.

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

Step 1.2.4.4.4.5.5

Evaluate the exponent.

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

Step 1.2.4.4.5

Simplify the numerator.

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Step 1.2.4.4.5.1

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

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

Step 1.2.4.4.5.2

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

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

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Set the denominator in $\frac{1}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 1.3.2

Solve for $x$ .

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Step 1.3.2.1

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

$x = \pm \sqrt{0}$

Step 1.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.4

Evaluate $x^{2} + \frac{1}{x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{\sqrt[3]{4}}{2}$ .

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Step 1.4.1.1

Substitute $\frac{\sqrt[3]{4}}{2}$ for $x$ .

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

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

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

$\frac{{\sqrt[3]{4}}^{2}}{2^{2}} + \frac{1}{\frac{\sqrt[3]{4}}{2}}$

Step 1.4.1.2.1.2

Simplify the numerator.

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Step 1.4.1.2.1.2.1

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

$\frac{\sqrt[3]{4^{2}}}{2^{2}} + \frac{1}{\frac{\sqrt[3]{4}}{2}}$

Step 1.4.1.2.1.2.2

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

$\frac{\sqrt[3]{16}}{2^{2}} + \frac{1}{\frac{\sqrt[3]{4}}{2}}$

Step 1.4.1.2.1.2.3

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

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Step 1.4.1.2.1.2.3.1

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{8 (2)}}{2^{2}} + \frac{1}{\frac{\sqrt[3]{4}}{2}}$

Step 1.4.1.2.1.2.3.2

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

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

Step 1.4.1.2.1.2.4

Pull terms out from under the radical.

$\frac{2 \sqrt[3]{2}}{2^{2}} + \frac{1}{\frac{\sqrt[3]{4}}{2}}$

Step 1.4.1.2.1.3

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

$\frac{2 \sqrt[3]{2}}{4} + \frac{1}{\frac{\sqrt[3]{4}}{2}}$

Step 1.4.1.2.1.4

Cancel the common factor of $2$ and $4$ .

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Step 1.4.1.2.1.4.1

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

$\frac{2 (\sqrt[3]{2})}{4} + \frac{1}{\frac{\sqrt[3]{4}}{2}}$

Step 1.4.1.2.1.4.2

Cancel the common factors.

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Step 1.4.1.2.1.4.2.1

Factor $2$ out of $4$ .

$\frac{2 \sqrt[3]{2}}{2 \cdot 2} + \frac{1}{\frac{\sqrt[3]{4}}{2}}$

Step 1.4.1.2.1.4.2.2

Cancel the common factor.

$\frac{2 \sqrt[3]{2}}{2 \cdot 2} + \frac{1}{\frac{\sqrt[3]{4}}{2}}$

Step 1.4.1.2.1.4.2.3

Rewrite the expression.

$\frac{\sqrt[3]{2}}{2} + \frac{1}{\frac{\sqrt[3]{4}}{2}}$

Step 1.4.1.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt[3]{2}}{2} + 1 \frac{2}{\sqrt[3]{4}}$

Step 1.4.1.2.1.6

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

$\frac{\sqrt[3]{2}}{2} + \frac{2}{\sqrt[3]{4}}$

Step 1.4.1.2.1.7

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

$\frac{\sqrt[3]{2}}{2} + \frac{2}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 1.4.1.2.1.8

Combine and simplify the denominator.

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Step 1.4.1.2.1.8.1

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

$\frac{\sqrt[3]{2}}{2} + \frac{2 {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 1.4.1.2.1.8.2

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

$\frac{\sqrt[3]{2}}{2} + \frac{2 {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 1.4.1.2.1.8.3

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

$\frac{\sqrt[3]{2}}{2} + \frac{2 {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 1.4.1.2.1.8.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{2}}{2} + \frac{2 {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 1.4.1.2.1.8.5

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

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Step 1.4.1.2.1.8.5.1

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

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

Step 1.4.1.2.1.8.5.2

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

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

Step 1.4.1.2.1.8.5.3

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

$\frac{\sqrt[3]{2}}{2} + \frac{2 {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 1.4.1.2.1.8.5.4

Cancel the common factor of $3$ .

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Step 1.4.1.2.1.8.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{2}}{2} + \frac{2 {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 1.4.1.2.1.8.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{2}}{2} + \frac{2 {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 1.4.1.2.1.8.5.5

Evaluate the exponent.

$\frac{\sqrt[3]{2}}{2} + \frac{2 {\sqrt[3]{4}}^{2}}{4}$

Step 1.4.1.2.1.9

Cancel the common factor of $2$ and $4$ .

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Step 1.4.1.2.1.9.1

Factor $2$ out of $2 {\sqrt[3]{4}}^{2}$ .

$\frac{\sqrt[3]{2}}{2} + \frac{2 ({\sqrt[3]{4}}^{2})}{4}$

Step 1.4.1.2.1.9.2

Cancel the common factors.

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Step 1.4.1.2.1.9.2.1

Factor $2$ out of $4$ .

$\frac{\sqrt[3]{2}}{2} + \frac{2 {\sqrt[3]{4}}^{2}}{2 \cdot 2}$

Step 1.4.1.2.1.9.2.2

Cancel the common factor.

$\frac{\sqrt[3]{2}}{2} + \frac{2 {\sqrt[3]{4}}^{2}}{2 \cdot 2}$

Step 1.4.1.2.1.9.2.3

Rewrite the expression.

$\frac{\sqrt[3]{2}}{2} + \frac{{\sqrt[3]{4}}^{2}}{2}$

Step 1.4.1.2.1.10

Simplify the numerator.

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Step 1.4.1.2.1.10.1

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

$\frac{\sqrt[3]{2}}{2} + \frac{\sqrt[3]{4^{2}}}{2}$

Step 1.4.1.2.1.10.2

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

$\frac{\sqrt[3]{2}}{2} + \frac{\sqrt[3]{16}}{2}$

Step 1.4.1.2.1.10.3

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

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Step 1.4.1.2.1.10.3.1

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{2}}{2} + \frac{\sqrt[3]{8 (2)}}{2}$

Step 1.4.1.2.1.10.3.2

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

$\frac{\sqrt[3]{2}}{2} + \frac{\sqrt[3]{2^{3} \cdot 2}}{2}$

Step 1.4.1.2.1.10.4

Pull terms out from under the radical.

$\frac{\sqrt[3]{2}}{2} + \frac{2 \sqrt[3]{2}}{2}$

Step 1.4.1.2.1.11

Cancel the common factor of $2$ .

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Step 1.4.1.2.1.11.1

Cancel the common factor.

$\frac{\sqrt[3]{2}}{2} + \frac{2 \sqrt[3]{2}}{2}$

Step 1.4.1.2.1.11.2

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

$\frac{\sqrt[3]{2}}{2} + \sqrt[3]{2}$

Step 1.4.1.2.2

To write $\sqrt[3]{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sqrt[3]{2}}{2} + \sqrt[3]{2} \cdot \frac{2}{2}$

Step 1.4.1.2.3

Combine fractions.

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Step 1.4.1.2.3.1

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

$\frac{\sqrt[3]{2}}{2} + \frac{\sqrt[3]{2} \cdot 2}{2}$

Step 1.4.1.2.3.2

Combine the numerators over the common denominator.

$\frac{\sqrt[3]{2} + \sqrt[3]{2} \cdot 2}{2}$

Step 1.4.1.2.4

Simplify the numerator.

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Step 1.4.1.2.4.1

Move $2$ to the left of $\sqrt[3]{2}$ .

$\frac{\sqrt[3]{2} + 2 \cdot \sqrt[3]{2}}{2}$

Step 1.4.1.2.4.2

Add $\sqrt[3]{2}$ and $2 \sqrt[3]{2}$ .

$\frac{3 \sqrt[3]{2}}{2}$

Step 1.4.2

Evaluate at $x = 0$ .

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Step 1.4.2.1

Substitute $0$ for $x$ .

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

Step 1.4.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 1.4.3

List all of the points.

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

Step 2

Use the first derivative test to determine which points can be maxima or minima.

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Step 2.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

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

Step 2.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, \frac{\sqrt[3]{4}}{2})$ in the first derivative $2 x - \frac{1}{x^{2}}$ to check if the result is negative or positive.

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Step 2.2.1

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

$f' (- 2) = 2 (- 2) - \frac{1}{{(- 2)}^{2}}$

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $2$ by $- 2$ .

$f' (- 2) = - 4 - \frac{1}{{(- 2)}^{2}}$

Step 2.2.2.1.2

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

$f' (- 2) = - 4 - \frac{1}{4}$

Step 2.2.2.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f' (- 2) = - 4 \cdot \frac{4}{4} - \frac{1}{4}$

Step 2.2.2.3

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

$f' (- 2) = \frac{- 4 \cdot 4}{4} - \frac{1}{4}$

Step 2.2.2.4

Combine the numerators over the common denominator.

$f' (- 2) = \frac{- 4 \cdot 4 - 1}{4}$

Step 2.2.2.5

Simplify the numerator.

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Step 2.2.2.5.1

Multiply $- 4$ by $4$ .

$f' (- 2) = \frac{- 16 - 1}{4}$

Step 2.2.2.5.2

Subtract $1$ from $- 16$ .

$f' (- 2) = \frac{- 17}{4}$

Step 2.2.2.6

Move the negative in front of the fraction.

$f' (- 2) = - \frac{17}{4}$

Step 2.2.2.7

The final answer is $- \frac{17}{4}$ .

$- \frac{17}{4}$

Step 2.3

Substitute any number, such as $3$ , from the interval $(\frac{\sqrt[3]{4}}{2}, \infty)$ in the first derivative $2 x - \frac{1}{x^{2}}$ to check if the result is negative or positive.

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Step 2.3.1

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

$f' (3) = 2 (3) - \frac{1}{{(3)}^{2}}$

Step 2.3.2

Simplify the result.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $2$ by $3$ .

$f' (3) = 6 - \frac{1}{{(3)}^{2}}$

Step 2.3.2.1.2

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

$f' (3) = 6 - \frac{1}{9}$

Step 2.3.2.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f' (3) = 6 \cdot \frac{9}{9} - \frac{1}{9}$

Step 2.3.2.3

Combine $6$ and $\frac{9}{9}$ .

$f' (3) = \frac{6 \cdot 9}{9} - \frac{1}{9}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$f' (3) = \frac{6 \cdot 9 - 1}{9}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $6$ by $9$ .

$f' (3) = \frac{54 - 1}{9}$

Step 2.3.2.5.2

Subtract $1$ from $54$ .

$f' (3) = \frac{53}{9}$

Step 2.3.2.6

The final answer is $\frac{53}{9}$ .

$\frac{53}{9}$

Step 2.4

Since the first derivative changed signs from negative to positive around $x = \frac{\sqrt[3]{4}}{2}$ , then $x = \frac{\sqrt[3]{4}}{2}$ is a local minimum.

$x = \frac{\sqrt[3]{4}}{2}$ is a local minimum

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

No absolute maximum

Absolute Minimum: $(\frac{\sqrt[3]{4}}{2}, \frac{3 \sqrt[3]{2}}{2})$

Step 4