Calculus Examples

$x - 4 \sqrt{x}$

Step 1

Write $x - 4 \sqrt{x}$ as a function.

$f (x) = x - 4 \sqrt{x}$

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x - 4 \sqrt{x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4 \sqrt{x}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 4 \sqrt{x}]$

Step 2.1.2

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

$1 + \frac{d}{d x} [- 4 \sqrt{x}]$

Step 2.2

Evaluate $\frac{d}{d x} [- 4 \sqrt{x}]$ .

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Step 2.2.1

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

$1 + \frac{d}{d x} [- 4 x^{\frac{1}{2}}]$

Step 2.2.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{\frac{1}{2}}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$1 - 4 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 2.2.3

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

$1 - 4 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 2.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$1 - 4 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 2.2.5

Combine $- 1$ and $\frac{2}{2}$ .

$1 - 4 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 2.2.6

Combine the numerators over the common denominator.

$1 - 4 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 2.2.7

Simplify the numerator.

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Step 2.2.7.1

Multiply $- 1$ by $2$ .

$1 - 4 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 2.2.7.2

Subtract $2$ from $1$ .

$1 - 4 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 2.2.8

Move the negative in front of the fraction.

$1 - 4 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 2.2.9

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

$1 - 4 \frac{x^{- \frac{1}{2}}}{2}$

Step 2.2.10

Combine $- 4$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$1 + \frac{- 4 x^{- \frac{1}{2}}}{2}$

Step 2.2.11

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 + \frac{- 4}{2 x^{\frac{1}{2}}}$

Step 2.2.12

Factor $2$ out of $- 4$ .

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

Step 2.2.13

Cancel the common factors.

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Step 2.2.13.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$1 + \frac{2 \cdot - 2}{2 (x^{\frac{1}{2}})}$

Step 2.2.13.2

Cancel the common factor.

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

Step 2.2.13.3

Rewrite the expression.

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

Step 2.2.14

Move the negative in front of the fraction.

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

Step 3

Find the second derivative of the function.

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Step 3.1

Differentiate.

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Step 3.1.1

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

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

Step 3.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} (- \frac{2}{x^{\frac{1}{2}}})$

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}}$ .

$f'' (x) = 0 - 2 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{\frac{1}{2}}))$

Step 3.2.3.2

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

$f'' (x) = 0 - 2 (- u^{- 2} \frac{d}{d x} x^{\frac{1}{2}})$

Step 3.2.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

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

Step 3.2.4

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

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

Step 3.2.5

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 2}$ .

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Step 3.2.5.1

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

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

Step 3.2.5.2

Cancel the common factor of $2$ .

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Step 3.2.5.2.1

Factor $2$ out of $- 2$ .

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

Step 3.2.5.2.2

Cancel the common factor.

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

Step 3.2.5.2.3

Rewrite the expression.

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

Step 3.2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.2.7

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (x) = 0 - 2 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 3.2.8

Combine the numerators over the common denominator.

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

Step 3.2.9

Simplify the numerator.

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Step 3.2.9.1

Multiply $- 1$ by $2$ .

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

Step 3.2.9.2

Subtract $2$ from $1$ .

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

Step 3.2.10

Move the negative in front of the fraction.

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

Step 3.2.11

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

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

Step 3.2.12

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

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

Step 3.2.13

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 3.2.13.1

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

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

Step 3.2.13.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.2.13.3

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (x) = 0 - 2 (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2})$

Step 3.2.13.4

Combine the numerators over the common denominator.

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

Step 3.2.13.5

Simplify the numerator.

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Step 3.2.13.5.1

Multiply $- 1$ by $2$ .

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

Step 3.2.13.5.2

Subtract $2$ from $- 1$ .

$f'' (x) = 0 - 2 (- \frac{x^{\frac{- 3}{2}}}{2})$

Step 3.2.13.6

Move the negative in front of the fraction.

$f'' (x) = 0 - 2 (- \frac{x^{- \frac{3}{2}}}{2})$

Step 3.2.14

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = 0 - 2 (- \frac{1}{2 x^{\frac{3}{2}}})$

Step 3.2.15

Multiply $- 1$ by $- 2$ .

$f'' (x) = 0 + 2 (\frac{1}{2 x^{\frac{3}{2}}})$

Step 3.2.16

Combine $2$ and $\frac{1}{2 x^{\frac{3}{2}}}$ .

$f'' (x) = 0 + \frac{2}{2 x^{\frac{3}{2}}}$

Step 3.2.17

Cancel the common factor.

$f'' (x) = 0 + \frac{2}{2 x^{\frac{3}{2}}}$

Step 3.2.18

Rewrite the expression.

$f'' (x) = 0 + \frac{1}{x^{\frac{3}{2}}}$

Step 3.3

Add $0$ and $\frac{1}{x^{\frac{3}{2}}}$ .

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

Step 4

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

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

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Differentiate.

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Step 5.1.1.1

By the Sum Rule, the derivative of $x - 4 \sqrt{x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4 \sqrt{x}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 4 \sqrt{x}]$

Step 5.1.1.2

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

$1 + \frac{d}{d x} [- 4 \sqrt{x}]$

Step 5.1.2

Evaluate $\frac{d}{d x} [- 4 \sqrt{x}]$ .

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Step 5.1.2.1

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

$1 + \frac{d}{d x} [- 4 x^{\frac{1}{2}}]$

Step 5.1.2.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{\frac{1}{2}}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$1 - 4 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 5.1.2.3

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

$1 - 4 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 5.1.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$1 - 4 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 5.1.2.5

Combine $- 1$ and $\frac{2}{2}$ .

$1 - 4 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 5.1.2.6

Combine the numerators over the common denominator.

$1 - 4 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 5.1.2.7

Simplify the numerator.

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Step 5.1.2.7.1

Multiply $- 1$ by $2$ .

$1 - 4 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 5.1.2.7.2

Subtract $2$ from $1$ .

$1 - 4 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 5.1.2.8

Move the negative in front of the fraction.

$1 - 4 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 5.1.2.9

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

$1 - 4 \frac{x^{- \frac{1}{2}}}{2}$

Step 5.1.2.10

Combine $- 4$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$1 + \frac{- 4 x^{- \frac{1}{2}}}{2}$

Step 5.1.2.11

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 + \frac{- 4}{2 x^{\frac{1}{2}}}$

Step 5.1.2.12

Factor $2$ out of $- 4$ .

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

Step 5.1.2.13

Cancel the common factors.

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Step 5.1.2.13.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$1 + \frac{2 \cdot - 2}{2 (x^{\frac{1}{2}})}$

Step 5.1.2.13.2

Cancel the common factor.

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

Step 5.1.2.13.3

Rewrite the expression.

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

Step 5.1.2.14

Move the negative in front of the fraction.

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Subtract $1$ from both sides of the equation.

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

Step 6.3

Find the LCD of the terms in the equation.

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Step 6.3.1

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

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

Step 6.3.2

The LCM of one and any expression is the expression.

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

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 6.4.2.1.1

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

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

Step 6.4.2.1.2

Cancel the common factor.

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

Step 6.4.2.1.3

Rewrite the expression.

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

Step 6.5

Solve the equation.

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Step 6.5.1

Rewrite the equation as $- x^{\frac{1}{2}} = - 2$ .

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

Step 6.5.2

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

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Step 6.5.2.1

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

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

Step 6.5.2.2

Simplify the left side.

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Step 6.5.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.5.2.2.2

Divide $x^{\frac{1}{2}}$ by $1$ .

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

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Divide $- 2$ by $- 1$ .

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

Step 6.5.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 6.5.4

Simplify the exponent.

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Step 6.5.4.1

Simplify the left side.

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Step 6.5.4.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 6.5.4.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 6.5.4.1.1.1.1

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

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

Step 6.5.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 6.5.4.1.1.1.2.1

Cancel the common factor.

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

Step 6.5.4.1.1.1.2.2

Rewrite the expression.

$x^{1} = 2^{2}$

Step 6.5.4.1.1.2

Simplify.

$x = 2^{2}$

Step 6.5.4.2

Simplify the right side.

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Step 6.5.4.2.1

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

$x = 4$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Convert expressions with fractional exponents to radicals.

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Step 7.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 7.1.2

Anything raised to $1$ is the base itself.

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

Step 7.2

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

$\sqrt{x} = 0$

Step 7.3

Solve for $x$ .

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Step 7.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 7.3.2

Simplify each side of the equation.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 7.3.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 7.3.2.2.1.1.1

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

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

Step 7.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 7.3.2.2.1.1.2.1

Cancel the common factor.

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

Step 7.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 7.3.2.2.1.2

Simplify.

$x = 0^{2}$

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 7.4

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 7.5

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 8

Critical points to evaluate.

$x = 4, 0$

Step 9

Evaluate the second derivative at $x = 4$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{1}{{(4)}^{\frac{3}{2}}}$

Step 10

Simplify the denominator.

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Step 10.1

Rewrite $4$ as $2^{2}$ .

$\frac{1}{{(2^{2})}^{\frac{3}{2}}}$

Step 10.2

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

$\frac{1}{2^{2 (\frac{3}{2})}}$

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

Cancel the common factor.

$\frac{1}{2^{2 (\frac{3}{2})}}$

Step 10.3.2

Rewrite the expression.

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

Step 10.4

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

$\frac{1}{8}$

Step 11

$x = 4$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 4$ is a local minimum

Step 12

Find the y-value when $x = 4$ .

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Step 12.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = (4) - 4 \sqrt{4}$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Rewrite $4$ as $2^{2}$ .

$f (4) = 4 - 4 \sqrt{2^{2}}$

Step 12.2.1.2

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

$f (4) = 4 - 4 \cdot 2$

Step 12.2.1.3

Multiply $- 4$ by $2$ .

$f (4) = 4 - 8$

Step 12.2.2

Subtract $8$ from $4$ .

$f (4) = - 4$

Step 12.2.3

The final answer is $- 4$ .

$y = - 4$

Step 13

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{1}{{(0)}^{\frac{3}{2}}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify the expression.

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Step 14.1.1

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

$\frac{1}{{(0^{2})}^{\frac{3}{2}}}$

Step 14.1.2

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

$\frac{1}{0^{2 (\frac{3}{2})}}$

Step 14.2

Cancel the common factor of $2$ .

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Step 14.2.1

Cancel the common factor.

$\frac{1}{0^{2 (\frac{3}{2})}}$

Step 14.2.2

Rewrite the expression.

$\frac{1}{0^{3}}$

Step 14.3

Raising $0$ to any positive power yields $0$ .

$\frac{1}{0}$

Step 14.4

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

Undefined

Step 15

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 16