Calculus Examples

$f (x) = 9 x^{\frac{2}{3}} - x$ on $0$ , $729$

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

By the Sum Rule, the derivative of $9 x^{\frac{2}{3}} - x$ with respect to $x$ is $\frac{d}{d x} [9 x^{\frac{2}{3}}] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [9 x^{\frac{2}{3}}] + \frac{d}{d x} [- x]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [9 x^{\frac{2}{3}}]$ .

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Step 1.1.1.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{\frac{2}{3}}$ with respect to $x$ is $9 \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$9 \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- x]$

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 $n = \frac{2}{3}$ .

$9 (\frac{2}{3} x^{\frac{2}{3} - 1}) + \frac{d}{d x} [- x]$

Step 1.1.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$9 (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [- x]$

Step 1.1.1.2.4

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

$9 (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [- x]$

Step 1.1.1.2.5

Combine the numerators over the common denominator.

$9 (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}) + \frac{d}{d x} [- x]$

Step 1.1.1.2.6

Simplify the numerator.

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Step 1.1.1.2.6.1

Multiply $- 1$ by $3$ .

$9 (\frac{2}{3} x^{\frac{2 - 3}{3}}) + \frac{d}{d x} [- x]$

Step 1.1.1.2.6.2

Subtract $3$ from $2$ .

$9 (\frac{2}{3} x^{\frac{- 1}{3}}) + \frac{d}{d x} [- x]$

Step 1.1.1.2.7

Move the negative in front of the fraction.

$9 (\frac{2}{3} x^{- \frac{1}{3}}) + \frac{d}{d x} [- x]$

Step 1.1.1.2.8

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

$9 \frac{2 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [- x]$

Step 1.1.1.2.9

Combine $9$ and $\frac{2 x^{- \frac{1}{3}}}{3}$ .

$\frac{9 (2 x^{- \frac{1}{3}})}{3} + \frac{d}{d x} [- x]$

Step 1.1.1.2.10

Multiply $2$ by $9$ .

$\frac{18 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [- x]$

Step 1.1.1.2.11

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

$\frac{18}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- x]$

Step 1.1.1.2.12

Factor $3$ out of $18$ .

$\frac{3 \cdot 6}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- x]$

Step 1.1.1.2.13

Cancel the common factors.

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Step 1.1.1.2.13.1

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

$\frac{3 \cdot 6}{3 (x^{\frac{1}{3}})} + \frac{d}{d x} [- x]$

Step 1.1.1.2.13.2

Cancel the common factor.

$\frac{3 \cdot 6}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- x]$

Step 1.1.1.2.13.3

Rewrite the expression.

$\frac{6}{x^{\frac{1}{3}}} + \frac{d}{d x} [- x]$

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$\frac{6}{x^{\frac{1}{3}}} - \frac{d}{d x} [x]$

Step 1.1.1.3.2

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

$\frac{6}{x^{\frac{1}{3}}} - 1 \cdot 1$

Step 1.1.1.3.3

Multiply $- 1$ by $1$ .

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

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{6}{x^{\frac{1}{3}}} - 1$ .

$\frac{6}{x^{\frac{1}{3}}} - 1$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{6}{x^{\frac{1}{3}}} - 1 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{6}{x^{\frac{1}{3}}} - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$\frac{6}{x^{\frac{1}{3}}} = 1$

Step 1.2.3

Find the LCD of the terms in the equation.

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Step 1.2.3.1

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

$x^{\frac{1}{3}}, 1$

Step 1.2.3.2

The LCM of one and any expression is the expression.

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

Step 1.2.4

Multiply each term in $\frac{6}{x^{\frac{1}{3}}} = 1$ by $x^{\frac{1}{3}}$ to eliminate the fractions.

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Step 1.2.4.1

Multiply each term in $\frac{6}{x^{\frac{1}{3}}} = 1$ by $x^{\frac{1}{3}}$ .

$\frac{6}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = 1 x^{\frac{1}{3}}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $x^{\frac{1}{3}}$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{6}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = 1 x^{\frac{1}{3}}$

Step 1.2.4.2.1.2

Rewrite the expression.

$6 = 1 x^{\frac{1}{3}}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Multiply $x^{\frac{1}{3}}$ by $1$ .

$6 = x^{\frac{1}{3}}$

Step 1.2.5

Solve the equation.

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Step 1.2.5.1

Rewrite the equation as $x^{\frac{1}{3}} = 6$ .

$x^{\frac{1}{3}} = 6$

Step 1.2.5.2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = 6^{3}$

Step 1.2.5.3

Simplify the exponent.

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Step 1.2.5.3.1

Simplify the left side.

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Step 1.2.5.3.1.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.5.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.5.3.1.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = 6^{3}$

Step 1.2.5.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.5.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 6^{3}$

Step 1.2.5.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 6^{3}$

Step 1.2.5.3.1.1.2

Simplify.

$x = 6^{3}$

Step 1.2.5.3.2

Simplify the right side.

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Step 1.2.5.3.2.1

Raise $6$ to the power of $3$ .

$x = 216$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Convert expressions with fractional exponents to radicals.

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Step 1.3.1.1

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

$\frac{6}{\sqrt[3]{x^{1}}} - 1$

Step 1.3.1.2

Anything raised to $1$ is the base itself.

$\frac{6}{\sqrt[3]{x}} - 1$

Step 1.3.2

Set the denominator in $\frac{6}{\sqrt[3]{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x} = 0$

Step 1.3.3

Solve for $x$ .

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Step 1.3.3.1

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

${\sqrt[3]{x}}^{3} = 0^{3}$

Step 1.3.3.2

Simplify each side of the equation.

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Step 1.3.3.2.1

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

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 1.3.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 1.3.3.2.2.1.2

Simplify.

$x = 0^{3}$

Step 1.3.3.2.3

Simplify the right side.

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Step 1.3.3.2.3.1

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

$x = 0$

Step 1.4

Evaluate $9 x^{\frac{2}{3}} - x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 216$ .

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Step 1.4.1.1

Substitute $216$ for $x$ .

$9 {(216)}^{\frac{2}{3}} - (216)$

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

Rewrite $216$ as $6^{3}$ .

$9 {(6^{3})}^{\frac{2}{3}} - (216)$

Step 1.4.1.2.1.2

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

$9 \cdot 6^{3 (\frac{2}{3})} - (216)$

Step 1.4.1.2.1.3

Cancel the common factor of $3$ .

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Step 1.4.1.2.1.3.1

Cancel the common factor.

$9 \cdot 6^{3 (\frac{2}{3})} - (216)$

Step 1.4.1.2.1.3.2

Rewrite the expression.

$9 \cdot 6^{2} - (216)$

Step 1.4.1.2.1.4

Raise $6$ to the power of $2$ .

$9 \cdot 36 - (216)$

Step 1.4.1.2.1.5

Multiply $9$ by $36$ .

$324 - (216)$

Step 1.4.1.2.1.6

Multiply $- 1$ by $216$ .

$324 - 216$

Step 1.4.1.2.2

Subtract $216$ from $324$ .

$108$

Step 1.4.2

Evaluate at $x = 0$ .

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Step 1.4.2.1

Substitute $0$ for $x$ .

$9 {(0)}^{\frac{2}{3}} - (0)$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify the expression.

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Step 1.4.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$9 {(0^{3})}^{\frac{2}{3}} - (0)$

Step 1.4.2.2.1.2

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

$9 \cdot 0^{3 (\frac{2}{3})} - (0)$

Step 1.4.2.2.2

Cancel the common factor of $3$ .

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Step 1.4.2.2.2.1

Cancel the common factor.

$9 \cdot 0^{3 (\frac{2}{3})} - (0)$

Step 1.4.2.2.2.2

Rewrite the expression.

$9 \cdot 0^{2} - (0)$

Step 1.4.2.2.3

Simplify the expression.

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Step 1.4.2.2.3.1

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

$9 \cdot 0 - (0)$

Step 1.4.2.2.3.2

Multiply $9$ by $0$ .

$0 - (0)$

Step 1.4.2.2.3.3

Subtract $0$ from $0$ .

$0$

Step 1.4.3

List all of the points.

$(216, 108), (0, 0)$

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 $x$ .

$9 {(0)}^{\frac{2}{3}} - (0)$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify the expression.

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Step 2.1.2.1.1

Rewrite $0$ as $0^{3}$ .

$9 {(0^{3})}^{\frac{2}{3}} - (0)$

Step 2.1.2.1.2

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

$9 \cdot 0^{3 (\frac{2}{3})} - (0)$

Step 2.1.2.2

Cancel the common factor of $3$ .

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Step 2.1.2.2.1

Cancel the common factor.

$9 \cdot 0^{3 (\frac{2}{3})} - (0)$

Step 2.1.2.2.2

Rewrite the expression.

$9 \cdot 0^{2} - (0)$

Step 2.1.2.3

Simplify the expression.

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Step 2.1.2.3.1

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

$9 \cdot 0 - (0)$

Step 2.1.2.3.2

Multiply $9$ by $0$ .

$0 - (0)$

Step 2.1.2.3.3

Subtract $0$ from $0$ .

$0$

Step 2.2

Evaluate at $x = 729$ .

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Step 2.2.1

Substitute $729$ for $x$ .

$9 {(729)}^{\frac{2}{3}} - (729)$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Rewrite $729$ as $9^{3}$ .

$9 {(9^{3})}^{\frac{2}{3}} - (729)$

Step 2.2.2.1.2

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

$9 \cdot 9^{3 (\frac{2}{3})} - (729)$

Step 2.2.2.1.3

Cancel the common factor of $3$ .

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Step 2.2.2.1.3.1

Cancel the common factor.

$9 \cdot 9^{3 (\frac{2}{3})} - (729)$

Step 2.2.2.1.3.2

Rewrite the expression.

$9 \cdot 9^{2} - (729)$

Step 2.2.2.1.4

Raise $9$ to the power of $2$ .

$9 \cdot 81 - (729)$

Step 2.2.2.1.5

Multiply $9$ by $81$ .

$729 - (729)$

Step 2.2.2.1.6

Multiply $- 1$ by $729$ .

$729 - 729$

Step 2.2.2.2

Subtract $729$ from $729$ .

$0$

Step 2.3

List all of the points.

$(0, 0), (729, 0)$

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: $(216, 108)$

Absolute Minimum: $(0, 0), (729, 0)$

Step 4