Calculus Examples

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

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

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

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

Step 1.1.1.2

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

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

Cancel the common factor of $2$ .

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Step 1.1.1.3.2.1

Cancel the common factor.

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

Step 1.1.1.3.2.2

Rewrite the expression.

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

Step 1.1.1.4

Simplify.

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

Step 1.1.1.5

Differentiate using the Power Rule.

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Step 1.1.1.5.1

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

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

Step 1.1.1.5.2

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

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

Step 1.1.1.6

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

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Step 1.1.1.6.1

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

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

Step 1.1.1.6.2

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

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

Step 1.1.1.6.3

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

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

Step 1.1.1.7

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

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

Step 1.1.1.8

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

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

Step 1.1.1.9

Combine the numerators over the common denominator.

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

Step 1.1.1.10

Simplify the numerator.

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Step 1.1.1.10.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.10.2

Subtract $2$ from $1$ .

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

Step 1.1.1.11

Combine fractions.

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Step 1.1.1.11.1

Move the negative in front of the fraction.

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

Step 1.1.1.11.2

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

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

Step 1.1.1.11.3

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

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

Step 1.1.1.11.4

Combine $\frac{1}{2 {(x^{2} + 1)}^{\frac{1}{2}}}$ and $x$ .

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

Step 1.1.1.12

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

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

Step 1.1.1.13

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

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

Step 1.1.1.14

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

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

Step 1.1.1.15

Combine fractions.

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Step 1.1.1.15.1

Add $2 x$ and $0$ .

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

Step 1.1.1.15.2

Multiply $2$ by $- 1$ .

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

Step 1.1.1.15.3

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

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

Step 1.1.1.15.4

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

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

Step 1.1.1.16

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

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

Step 1.1.1.17

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

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

Step 1.1.1.18

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

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

Step 1.1.1.19

Add $1$ and $1$ .

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

Step 1.1.1.20

Factor $2$ out of $- 2 x^{2}$ .

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

Step 1.1.1.21

Cancel the common factors.

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Step 1.1.1.21.1

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

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

Step 1.1.1.21.2

Cancel the common factor.

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

Step 1.1.1.21.3

Rewrite the expression.

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

Step 1.1.1.22

Move the negative in front of the fraction.

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

Step 1.1.1.23

To write ${(x^{2} + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x^{2} + 1)}^{\frac{1}{2}}}{{(x^{2} + 1)}^{\frac{1}{2}}}$ .

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

Step 1.1.1.24

Combine the numerators over the common denominator.

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

Step 1.1.1.25

Multiply ${(x^{2} + 1)}^{\frac{1}{2}}$ by ${(x^{2} + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.1.1.25.1

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

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

Step 1.1.1.25.2

Combine the numerators over the common denominator.

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

Step 1.1.1.25.3

Add $1$ and $1$ .

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

Step 1.1.1.25.4

Divide $2$ by $2$ .

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

Step 1.1.1.26

Simplify ${(x^{2} + 1)}^{1}$ .

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

Step 1.1.1.27

Subtract $x^{2}$ from $x^{2}$ .

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

Step 1.1.1.28

Add $0$ and $1$ .

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

Step 1.1.1.29

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

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

Step 1.1.1.30

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

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

Step 1.1.1.31

Multiply ${(x^{2} + 1)}^{\frac{1}{2}}$ by $x^{2} + 1$ by adding the exponents.

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Step 1.1.1.31.1

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

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Step 1.1.1.31.1.1

Raise $x^{2} + 1$ to the power of $1$ .

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

Step 1.1.1.31.1.2

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

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

Step 1.1.1.31.2

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

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

Step 1.1.1.31.3

Combine the numerators over the common denominator.

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

Step 1.1.1.31.4

Add $1$ and $2$ .

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

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{1}{{(x^{2} + 1)}^{\frac{3}{2}}}$ .

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

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{1}{{(x^{2} + 1)}^{\frac{3}{2}}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$1 = 0$

Step 1.2.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 1.4

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = 0$ .

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Step 2.1.1

Substitute $0$ for $x$ .

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

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify the denominator.

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Step 2.1.2.1.1

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

$\frac{0}{\sqrt{0 + 1}}$

Step 2.1.2.1.2

Add $0$ and $1$ .

$\frac{0}{\sqrt{1}}$

Step 2.1.2.1.3

Any root of $1$ is $1$ .

$\frac{0}{1}$

Step 2.1.2.2

Divide $0$ by $1$ .

$0$

Step 2.2

Evaluate at $x = 2$ .

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Step 2.2.1

Substitute $2$ for $x$ .

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify the denominator.

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Step 2.2.2.1.1

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

$\frac{2}{\sqrt{4 + 1}}$

Step 2.2.2.1.2

Add $4$ and $1$ .

$\frac{2}{\sqrt{5}}$

Step 2.2.2.2

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 2.2.2.3

Combine and simplify the denominator.

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Step 2.2.2.3.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 2.2.2.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 2.2.2.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 2.2.2.3.4

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

$\frac{2 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 2.2.2.3.5

Add $1$ and $1$ .

$\frac{2 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 2.2.2.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.2.2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{2 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 2.2.2.3.6.2

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

$\frac{2 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 2.2.2.3.6.3

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

$\frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.2.2.3.6.4

Cancel the common factor of $2$ .

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Step 2.2.2.3.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.2.2.3.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{5}}{5^{1}}$

Step 2.2.2.3.6.5

Evaluate the exponent.

$\frac{2 \sqrt{5}}{5}$

Step 2.3

List all of the points.

$(0, 0), (2, \frac{2 \sqrt{5}}{5})$

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.

Absolute Maximum: $(2, \frac{2 \sqrt{5}}{5})$

Absolute Minimum: $(0, 0)$

Step 4