Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Move the negative in front of the fraction.

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

Factor $2$ out of $- 2$ .

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

Step 1.13

Cancel the common factors.

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Step 1.13.1

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

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

Step 1.13.2

Cancel the common factor.

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

Step 1.13.3

Rewrite the expression.

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

Step 1.14

Move the negative in front of the fraction.

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

Step 1.15

Simplify.

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Step 1.15.1

Apply the distributive property.

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

Step 1.15.2

Multiply $- 2$ by $2$ .

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

Step 1.15.3

Reorder the factors of $(2 x - 4 x^{\frac{1}{2}}) (1 - \frac{1}{x^{\frac{1}{2}}})$ .

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

Step 1.15.4

Expand $(1 - \frac{1}{x^{\frac{1}{2}}}) (2 x - 4 x^{\frac{1}{2}})$ using the FOIL Method.

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Step 1.15.4.1

Apply the distributive property.

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

Step 1.15.4.2

Apply the distributive property.

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

Step 1.15.4.3

Apply the distributive property.

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

Step 1.15.5

Simplify and combine like terms.

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Step 1.15.5.1

Simplify each term.

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Step 1.15.5.1.1

Multiply $2 x$ by $1$ .

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

Step 1.15.5.1.2

Multiply $- 4 x^{\frac{1}{2}}$ by $1$ .

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

Step 1.15.5.1.3

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

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Step 1.15.5.1.3.1

Multiply $2$ by $- 1$ .

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

Step 1.15.5.1.3.2

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

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

Step 1.15.5.1.3.3

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

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

Step 1.15.5.1.4

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

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

Step 1.15.5.1.5

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

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Step 1.15.5.1.5.1

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

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

Step 1.15.5.1.5.2

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

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Step 1.15.5.1.5.2.1

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

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

Step 1.15.5.1.5.2.2

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

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

Step 1.15.5.1.5.3

Write $1$ as a fraction with a common denominator.

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

Step 1.15.5.1.5.4

Combine the numerators over the common denominator.

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

Step 1.15.5.1.5.5

Add $- 1$ and $2$ .

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

Step 1.15.5.1.6

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

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Step 1.15.5.1.6.1

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

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

Step 1.15.5.1.6.2

Factor $x^{\frac{1}{2}}$ out of $- 4 x^{\frac{1}{2}}$ .

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

Step 1.15.5.1.6.3

Cancel the common factor.

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

Step 1.15.5.1.6.4

Rewrite the expression.

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

Step 1.15.5.1.7

Multiply $- 1$ by $- 4$ .

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

Step 1.15.5.2

Subtract $2 x^{\frac{1}{2}}$ from $- 4 x^{\frac{1}{2}}$ .

$2 x - 6 x^{\frac{1}{2}} + 4$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $2 x - 6 x^{\frac{1}{2}} + 4$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 6 x^{\frac{1}{2}}] + \frac{d}{d x} [4]$ .

$f'' (x) = \frac{d}{d x} (2 x) + \frac{d}{d x} (- 6 x^{\frac{1}{2}}) + \frac{d}{d x} (4)$

Step 2.2

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

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Step 2.2.1

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

$f'' (x) = 2 \frac{d}{d x} (x) + \frac{d}{d x} (- 6 x^{\frac{1}{2}}) + \frac{d}{d x} (4)$

Step 2.2.2

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

$f'' (x) = 2 \cdot 1 + \frac{d}{d x} (- 6 x^{\frac{1}{2}}) + \frac{d}{d x} (4)$

Step 2.2.3

Multiply $2$ by $1$ .

$f'' (x) = 2 + \frac{d}{d x} (- 6 x^{\frac{1}{2}}) + \frac{d}{d x} (4)$

Step 2.3

Evaluate $\frac{d}{d x} [- 6 x^{\frac{1}{2}}]$ .

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Step 2.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{\frac{1}{2}}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$f'' (x) = 2 - 6 \frac{d}{d x} x^{\frac{1}{2}} + \frac{d}{d x} (4)$

Step 2.3.2

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

$f'' (x) = 2 - 6 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} (4)$

Step 2.3.3

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

$f'' (x) = 2 - 6 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} (4)$

Step 2.3.4

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

$f'' (x) = 2 - 6 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} (4)$

Step 2.3.5

Combine the numerators over the common denominator.

$f'' (x) = 2 - 6 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} (4)$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Multiply $- 1$ by $2$ .

$f'' (x) = 2 - 6 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} (4)$

Step 2.3.6.2

Subtract $2$ from $1$ .

$f'' (x) = 2 - 6 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} (4)$

Step 2.3.7

Move the negative in front of the fraction.

$f'' (x) = 2 - 6 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} (4)$

Step 2.3.8

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

$f'' (x) = 2 - 6 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} (4)$

Step 2.3.9

Combine $- 6$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$f'' (x) = 2 + \frac{- 6 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} (4)$

Step 2.3.10

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

$f'' (x) = 2 + \frac{- 6}{2 x^{\frac{1}{2}}} + \frac{d}{d x} (4)$

Step 2.3.11

Factor $2$ out of $- 6$ .

$f'' (x) = 2 + \frac{2 \cdot - 3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} (4)$

Step 2.3.12

Cancel the common factors.

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Step 2.3.12.1

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

$f'' (x) = 2 + \frac{2 \cdot - 3}{2 (x^{\frac{1}{2}})} + \frac{d}{d x} (4)$

Step 2.3.12.2

Cancel the common factor.

$f'' (x) = 2 + \frac{2 \cdot - 3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} (4)$

Step 2.3.12.3

Rewrite the expression.

$f'' (x) = 2 + \frac{- 3}{x^{\frac{1}{2}}} + \frac{d}{d x} (4)$

Step 2.3.13

Move the negative in front of the fraction.

$f'' (x) = 2 - \frac{3}{x^{\frac{1}{2}}} + \frac{d}{d x} (4)$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 2.4.2

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

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

Step 3

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

$2 x - 6 x^{\frac{1}{2}} + 4 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.3

Differentiate.

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.3.4

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

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

Step 4.1.4

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

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

Step 4.1.5

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

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

Step 4.1.6

Combine the numerators over the common denominator.

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

Step 4.1.7

Simplify the numerator.

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Step 4.1.7.1

Multiply $- 1$ by $2$ .

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

Step 4.1.7.2

Subtract $2$ from $1$ .

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

Step 4.1.8

Move the negative in front of the fraction.

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

Step 4.1.9

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

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

Step 4.1.10

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

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

Step 4.1.11

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

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

Step 4.1.12

Factor $2$ out of $- 2$ .

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

Step 4.1.13

Cancel the common factors.

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Step 4.1.13.1

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

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

Step 4.1.13.2

Cancel the common factor.

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

Step 4.1.13.3

Rewrite the expression.

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

Step 4.1.14

Move the negative in front of the fraction.

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

Step 4.1.15

Simplify.

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Step 4.1.15.1

Apply the distributive property.

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

Step 4.1.15.2

Multiply $- 2$ by $2$ .

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

Step 4.1.15.3

Reorder the factors of $(2 x - 4 x^{\frac{1}{2}}) (1 - \frac{1}{x^{\frac{1}{2}}})$ .

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

Step 4.1.15.4

Expand $(1 - \frac{1}{x^{\frac{1}{2}}}) (2 x - 4 x^{\frac{1}{2}})$ using the FOIL Method.

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Step 4.1.15.4.1

Apply the distributive property.

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

Step 4.1.15.4.2

Apply the distributive property.

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

Step 4.1.15.4.3

Apply the distributive property.

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

Step 4.1.15.5

Simplify and combine like terms.

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Step 4.1.15.5.1

Simplify each term.

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Step 4.1.15.5.1.1

Multiply $2 x$ by $1$ .

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

Step 4.1.15.5.1.2

Multiply $- 4 x^{\frac{1}{2}}$ by $1$ .

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

Step 4.1.15.5.1.3

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

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Step 4.1.15.5.1.3.1

Multiply $2$ by $- 1$ .

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

Step 4.1.15.5.1.3.2

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

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

Step 4.1.15.5.1.3.3

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

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

Step 4.1.15.5.1.4

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

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

Step 4.1.15.5.1.5

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

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Step 4.1.15.5.1.5.1

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

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

Step 4.1.15.5.1.5.2

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

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Step 4.1.15.5.1.5.2.1

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

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

Step 4.1.15.5.1.5.2.2

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

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

Step 4.1.15.5.1.5.3

Write $1$ as a fraction with a common denominator.

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

Step 4.1.15.5.1.5.4

Combine the numerators over the common denominator.

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

Step 4.1.15.5.1.5.5

Add $- 1$ and $2$ .

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

Step 4.1.15.5.1.6

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

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Step 4.1.15.5.1.6.1

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

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

Step 4.1.15.5.1.6.2

Factor $x^{\frac{1}{2}}$ out of $- 4 x^{\frac{1}{2}}$ .

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

Step 4.1.15.5.1.6.3

Cancel the common factor.

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

Step 4.1.15.5.1.6.4

Rewrite the expression.

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

Step 4.1.15.5.1.7

Multiply $- 1$ by $- 4$ .

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

Step 4.1.15.5.2

Subtract $2 x^{\frac{1}{2}}$ from $- 4 x^{\frac{1}{2}}$ .

$f' (x) = 2 x - 6 x^{\frac{1}{2}} + 4$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $2 x - 6 x^{\frac{1}{2}} + 4$ .

$2 x - 6 x^{\frac{1}{2}} + 4$

Step 5

Set the first derivative equal to $0$ then solve the equation $2 x - 6 x^{\frac{1}{2}} + 4 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$2 x - 6 x^{\frac{1}{2}} + 4 = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

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

$2 {(x^{\frac{1}{2}})}^{2} - 6 x^{\frac{1}{2}} + 4 = 0$

Step 5.2.2

Let $u = x^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{2}}$ .

$2 u^{2} - 6 u + 4 = 0$

Step 5.2.3

Factor $2$ out of $2 u^{2} - 6 u + 4$ .

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Step 5.2.3.1

Factor $2$ out of $2 u^{2}$ .

$2 (u^{2}) - 6 u + 4 = 0$

Step 5.2.3.2

Factor $2$ out of $- 6 u$ .

$2 (u^{2}) + 2 (- 3 u) + 4 = 0$

Step 5.2.3.3

Factor $2$ out of $4$ .

$2 u^{2} + 2 (- 3 u) + 2 \cdot 2 = 0$

Step 5.2.3.4

Factor $2$ out of $2 u^{2} + 2 (- 3 u)$ .

$2 (u^{2} - 3 u) + 2 \cdot 2 = 0$

Step 5.2.3.5

Factor $2$ out of $2 (u^{2} - 3 u) + 2 \cdot 2$ .

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

Step 5.2.4

Factor.

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Step 5.2.4.1

Factor $u^{2} - 3 u + 2$ using the AC method.

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Step 5.2.4.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 5.2.4.1.2

Write the factored form using these integers.

$2 ((u - 2) (u - 1)) = 0$

Step 5.2.4.2

Remove unnecessary parentheses.

$2 (u - 2) (u - 1) = 0$

Step 5.2.5

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

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

Step 5.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

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

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

Step 5.4

Set $x^{\frac{1}{2}} - 2$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set $x^{\frac{1}{2}} - 2$ equal to $0$ .

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

Step 5.4.2

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

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Step 5.4.2.1

Add $2$ to both sides of the equation.

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

Step 5.4.2.2

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 5.4.2.3

Simplify the exponent.

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Step 5.4.2.3.1

Simplify the left side.

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Step 5.4.2.3.1.1

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

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Step 5.4.2.3.1.1.1

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

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Step 5.4.2.3.1.1.1.1

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

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

Step 5.4.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.4.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 5.4.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 5.4.2.3.1.1.2

Simplify.

$x = 2^{2}$

Step 5.4.2.3.2

Simplify the right side.

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Step 5.4.2.3.2.1

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

$x = 4$

Step 5.5

Set $x^{\frac{1}{2}} - 1$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $x^{\frac{1}{2}} - 1$ equal to $0$ .

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

Step 5.5.2

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

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Step 5.5.2.1

Add $1$ to both sides of the equation.

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

Step 5.5.2.2

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

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

Step 5.5.2.3

Simplify the exponent.

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Step 5.5.2.3.1

Simplify the left side.

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Step 5.5.2.3.1.1

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

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Step 5.5.2.3.1.1.1

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

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Step 5.5.2.3.1.1.1.1

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

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

Step 5.5.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.5.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 5.5.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 5.5.2.3.1.1.2

Simplify.

$x = 1^{2}$

Step 5.5.2.3.2

Simplify the right side.

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Step 5.5.2.3.2.1

One to any power is one.

$x = 1$

Step 5.6

The final solution is all the values that make $2 (x^{\frac{1}{2}} - 2) (x^{\frac{1}{2}} - 1) = 0$ true.

$x = 4, 1$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

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

$2 x - 6 \sqrt{x^{1}} + 4$

Step 6.1.2

Anything raised to $1$ is the base itself.

$2 x - 6 \sqrt{x} + 4$

Step 6.2

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

$x < 0$

Step 6.3

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 $0$ .

$x < 0$

$(- \infty, 0)$

$x < 0$

$(- \infty, 0)$

Step 7

Critical points to evaluate.

$x = 4, 1$

Step 8

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.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

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

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

Step 9.1.2

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

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

Step 9.1.3

Cancel the common factor of $2$ .

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Step 9.1.3.1

Cancel the common factor.

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

Step 9.1.3.2

Rewrite the expression.

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

Step 9.1.4

Evaluate the exponent.

$2 - \frac{3}{2}$

Step 9.2

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

$2 \cdot \frac{2}{2} - \frac{3}{2}$

Step 9.3

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

$\frac{2 \cdot 2}{2} - \frac{3}{2}$

Step 9.4

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 - 3}{2}$

Step 9.5

Simplify the numerator.

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Step 9.5.1

Multiply $2$ by $2$ .

$\frac{4 - 3}{2}$

Step 9.5.2

Subtract $3$ from $4$ .

$\frac{1}{2}$

Step 10

$x = 4$ 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 11

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

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Step 11.1

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

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

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

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

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

Step 11.2.1.2

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

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

Step 11.2.1.3

Multiply $- 2$ by $2$ .

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

Step 11.2.2

Simplify the expression.

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Step 11.2.2.1

Subtract $4$ from $4$ .

$f (4) = 0^{2}$

Step 11.2.2.2

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

$f (4) = 0$

Step 11.2.3

The final answer is $0$ .

$y = 0$

Step 12

Evaluate the second derivative at $x = 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

One to any power is one.

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

Step 13.1.2

Divide $3$ by $1$ .

$2 - 1 \cdot 3$

Step 13.1.3

Multiply $- 1$ by $3$ .

$2 - 3$

Step 13.2

Subtract $3$ from $2$ .

$- 1$

Step 14

$x = 1$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 1$ is a local maximum

Step 15

Find the y-value when $x = 1$ .

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Step 15.1

Replace the variable $x$ with $1$ in the expression.

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

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Any root of $1$ is $1$ .

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

Step 15.2.1.2

Multiply $- 2$ by $1$ .

$f (1) = {(1 - 2)}^{2}$

Step 15.2.2

Simplify the expression.

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Step 15.2.2.1

Subtract $2$ from $1$ .

$f (1) = {(- 1)}^{2}$

Step 15.2.2.2

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

$f (1) = 1$

Step 15.2.3

The final answer is $1$ .

$y = 1$

Step 16

These are the local extrema for $f (x) = {(x - 2 \sqrt{x})}^{2}$ .

$(4, 0)$ is a local minima

$(1, 1)$ is a local maxima

Step 17