Calculus Examples

$y = x^{\frac{4}{3}} - x^{\frac{1}{3}}$

Step 1

Write $y = x^{\frac{4}{3}} - x^{\frac{1}{3}}$ as a function.

$f (x) = x^{\frac{4}{3}} - x^{\frac{1}{3}}$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

Tap for more steps...

Step 2.1

Find the second derivative.

Tap for more steps...

Step 2.1.1

Find the first derivative.

Tap for more steps...

Step 2.1.1.1

By the Sum Rule, the derivative of $x^{\frac{4}{3}} - x^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{4}{3}}] + \frac{d}{d x} [- x^{\frac{1}{3}}]$ .

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

Step 2.1.1.2

Evaluate $\frac{d}{d x} [x^{\frac{4}{3}}]$ .

Tap for more steps...

Step 2.1.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

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

Step 2.1.1.2.2

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

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

Step 2.1.1.2.3

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

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

Step 2.1.1.2.4

Combine the numerators over the common denominator.

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

Step 2.1.1.2.5

Simplify the numerator.

Tap for more steps...

Step 2.1.1.2.5.1

Multiply $- 1$ by $3$ .

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

Step 2.1.1.2.5.2

Subtract $3$ from $4$ .

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

Step 2.1.1.3

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

Tap for more steps...

Step 2.1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{1}{3}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{1}{3}}]$ .

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

Step 2.1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{3}$ .

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

Step 2.1.1.3.3

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

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

Step 2.1.1.3.4

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

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

Step 2.1.1.3.5

Combine the numerators over the common denominator.

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

Step 2.1.1.3.6

Simplify the numerator.

Tap for more steps...

Step 2.1.1.3.6.1

Multiply $- 1$ by $3$ .

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

Step 2.1.1.3.6.2

Subtract $3$ from $1$ .

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

Step 2.1.1.3.7

Move the negative in front of the fraction.

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

Step 2.1.1.3.8

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

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

Step 2.1.1.3.9

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

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

Step 2.1.1.4

Combine $\frac{4}{3}$ and $x^{\frac{1}{3}}$ .

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

Step 2.1.2

Find the second derivative.

Tap for more steps...

Step 2.1.2.1

By the Sum Rule, the derivative of $\frac{4 x^{\frac{1}{3}}}{3} - \frac{1}{3 x^{\frac{2}{3}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{4 x^{\frac{1}{3}}}{3}] + \frac{d}{d x} [- \frac{1}{3 x^{\frac{2}{3}}}]$ .

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

Step 2.1.2.2

Evaluate $\frac{d}{d x} [\frac{4 x^{\frac{1}{3}}}{3}]$ .

Tap for more steps...

Step 2.1.2.2.1

Since $\frac{4}{3}$ is constant with respect to $x$ , the derivative of $\frac{4 x^{\frac{1}{3}}}{3}$ with respect to $x$ is $\frac{4}{3} \frac{d}{d x} [x^{\frac{1}{3}}]$ .

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

Step 2.1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{3}$ .

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

Step 2.1.2.2.3

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

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

Step 2.1.2.2.4

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

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

Step 2.1.2.2.5

Combine the numerators over the common denominator.

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

Step 2.1.2.2.6

Simplify the numerator.

Tap for more steps...

Step 2.1.2.2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.1.2.2.6.2

Subtract $3$ from $1$ .

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

Step 2.1.2.2.7

Move the negative in front of the fraction.

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

Step 2.1.2.2.8

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

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

Step 2.1.2.2.9

Multiply $\frac{4}{3}$ by $\frac{x^{- \frac{2}{3}}}{3}$ .

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

Step 2.1.2.2.10

Multiply $3$ by $3$ .

$f'' (x) = \frac{4 x^{- \frac{2}{3}}}{9} + \frac{d}{d x} (- \frac{1}{3 x^{\frac{2}{3}}})$

Step 2.1.2.2.11

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

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} + \frac{d}{d x} (- \frac{1}{3 x^{\frac{2}{3}}})$

Step 2.1.2.3

Evaluate $\frac{d}{d x} [- \frac{1}{3 x^{\frac{2}{3}}}]$ .

Tap for more steps...

Step 2.1.2.3.1

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{1}{3 x^{\frac{2}{3}}}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}]$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot \frac{d}{d x} \frac{1}{x^{\frac{2}{3}}}$

Step 2.1.2.3.2

Rewrite $\frac{1}{x^{\frac{2}{3}}}$ as ${(x^{\frac{2}{3}})}^{- 1}$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot \frac{d}{d x} {(x^{\frac{2}{3}})}^{- 1}$

Step 2.1.2.3.3

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 $g (x) = x^{\frac{2}{3}}$ .

Tap for more steps...

Step 2.1.2.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{2}{3}}$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{\frac{2}{3}}))$

Step 2.1.2.3.3.2

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

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- u^{- 2} \frac{d}{d x} x^{\frac{2}{3}})$

Step 2.1.2.3.3.3

Replace all occurrences of $u$ with $x^{\frac{2}{3}}$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- {(x^{\frac{2}{3}})}^{- 2} \frac{d}{d x} x^{\frac{2}{3}})$

Step 2.1.2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- {(x^{\frac{2}{3}})}^{- 2} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 2.1.2.3.5

Multiply the exponents in ${(x^{\frac{2}{3}})}^{- 2}$ .

Tap for more steps...

Step 2.1.2.3.5.1

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

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- x^{\frac{2}{3} \cdot - 2} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 2.1.2.3.5.2

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

Tap for more steps...

Step 2.1.2.3.5.2.1

Combine $\frac{2}{3}$ and $- 2$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- x^{\frac{2 \cdot - 2}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 2.1.2.3.5.2.2

Multiply $2$ by $- 2$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- x^{\frac{- 4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 2.1.2.3.5.3

Move the negative in front of the fraction.

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 2.1.2.3.6

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

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}))$

Step 2.1.2.3.7

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

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 2.1.2.3.8

Combine the numerators over the common denominator.

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}))$

Step 2.1.2.3.9

Simplify the numerator.

Tap for more steps...

Step 2.1.2.3.9.1

Multiply $- 1$ by $3$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 3}{3}}))$

Step 2.1.2.3.9.2

Subtract $3$ from $2$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{- 1}{3}}))$

Step 2.1.2.3.10

Move the negative in front of the fraction.

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- x^{- \frac{4}{3}} (\frac{2}{3} x^{- \frac{1}{3}}))$

Step 2.1.2.3.11

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

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- x^{- \frac{4}{3}} \frac{2 x^{- \frac{1}{3}}}{3})$

Step 2.1.2.3.12

Combine $\frac{2 x^{- \frac{1}{3}}}{3}$ and $x^{- \frac{4}{3}}$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- \frac{2 x^{- \frac{1}{3}} x^{- \frac{4}{3}}}{3})$

Step 2.1.2.3.13

Multiply $x^{- \frac{1}{3}}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

Tap for more steps...

Step 2.1.2.3.13.1

Move $x^{- \frac{4}{3}}$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- \frac{2 (x^{- \frac{4}{3}} x^{- \frac{1}{3}})}{3})$

Step 2.1.2.3.13.2

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

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- \frac{2 x^{- \frac{4}{3} - \frac{1}{3}}}{3})$

Step 2.1.2.3.13.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- \frac{2 x^{\frac{- 4 - 1}{3}}}{3})$

Step 2.1.2.3.13.4

Subtract $1$ from $- 4$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- \frac{2 x^{\frac{- 5}{3}}}{3})$

Step 2.1.2.3.13.5

Move the negative in front of the fraction.

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- \frac{2 x^{- \frac{5}{3}}}{3})$

Step 2.1.2.3.14

Move $x^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{1}{3} \cdot (- \frac{2}{3 x^{\frac{5}{3}}})$

Step 2.1.2.3.15

Multiply $- 1$ by $- 1$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} + 1 (\frac{1}{3}) \cdot \frac{2}{3 x^{\frac{5}{3}}}$

Step 2.1.2.3.16

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

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} + \frac{1}{3} \cdot \frac{2}{3 x^{\frac{5}{3}}}$

Step 2.1.2.3.17

Multiply $\frac{1}{3}$ by $\frac{2}{3 x^{\frac{5}{3}}}$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} + \frac{2}{3 (3 x^{\frac{5}{3}})}$

Step 2.1.2.3.18

Multiply $3$ by $3$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} + \frac{2}{9 x^{\frac{5}{3}}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{4}{9 x^{\frac{2}{3}}} + \frac{2}{9 x^{\frac{5}{3}}}$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{2}{9 x^{\frac{5}{3}}}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{4}{9 x^{\frac{2}{3}}} + \frac{2}{9 x^{\frac{5}{3}}} = 0$ .

Tap for more steps...

Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{2}{9 x^{\frac{5}{3}}} = 0$

Step 2.2.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.2.2.1

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

$9 x^{\frac{2}{3}}, 9 x^{\frac{5}{3}}, 1$

Step 2.2.2.2

Since $9 x^{\frac{2}{3}}, 9 x^{\frac{5}{3}}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $9, 9, 1$ then find LCM for the variable part $x^{\frac{2}{3}}, x^{\frac{5}{3}}$ .

Step 2.2.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.2.2.4

$9$ has factors of $3$ and $3$ .

$3 \cdot 3$

Step 2.2.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.2.2.6

The LCM of $9, 9, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3 \cdot 3$

Step 2.2.2.7

Multiply $3$ by $3$ .

$9$

Step 2.2.2.8

The LCM of $x^{\frac{2}{3}}, x^{\frac{5}{3}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x^{\frac{5}{3}}$

Step 2.2.2.9

The LCM for $9 x^{\frac{2}{3}}, 9 x^{\frac{5}{3}}, 1$ is the numeric part $9$ multiplied by the variable part.

$9 x^{\frac{5}{3}}$

Step 2.2.3

Multiply each term in $\frac{4}{9 x^{\frac{2}{3}}} + \frac{2}{9 x^{\frac{5}{3}}} = 0$ by $9 x^{\frac{5}{3}}$ to eliminate the fractions.

Tap for more steps...

Step 2.2.3.1

Multiply each term in $\frac{4}{9 x^{\frac{2}{3}}} + \frac{2}{9 x^{\frac{5}{3}}} = 0$ by $9 x^{\frac{5}{3}}$ .

$\frac{4}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) + \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2

Simplify the left side.

Tap for more steps...

Step 2.2.3.2.1

Simplify each term.

Tap for more steps...

Step 2.2.3.2.1.1

Rewrite using the commutative property of multiplication.

$9 \frac{4}{9 x^{\frac{2}{3}}} x^{\frac{5}{3}} + \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.2

Cancel the common factor of $9$ .

Tap for more steps...

Step 2.2.3.2.1.2.1

Cancel the common factor.

$9 \frac{4}{9 x^{\frac{2}{3}}} x^{\frac{5}{3}} + \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.2.2

Rewrite the expression.

$\frac{4}{x^{\frac{2}{3}}} x^{\frac{5}{3}} + \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.3

Cancel the common factor of $x^{\frac{2}{3}}$ .

Tap for more steps...

Step 2.2.3.2.1.3.1

Factor $x^{\frac{2}{3}}$ out of $x^{\frac{5}{3}}$ .

$\frac{4}{x^{\frac{2}{3}}} (x^{\frac{2}{3}} x^{\frac{3}{3}}) + \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.3.2

Cancel the common factor.

$\frac{4}{x^{\frac{2}{3}}} (x^{\frac{2}{3}} x^{\frac{3}{3}}) + \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.3.3

Rewrite the expression.

$4 x^{\frac{3}{3}} + \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.4

Divide $3$ by $3$ .

$4 x^{1} + \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.5

Simplify.

$4 x + \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.6

Rewrite using the commutative property of multiplication.

$4 x + 9 \frac{2}{9 x^{\frac{5}{3}}} x^{\frac{5}{3}} = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.7

Cancel the common factor of $9$ .

Tap for more steps...

Step 2.2.3.2.1.7.1

Cancel the common factor.

$4 x + 9 \frac{2}{9 x^{\frac{5}{3}}} x^{\frac{5}{3}} = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.7.2

Rewrite the expression.

$4 x + \frac{2}{x^{\frac{5}{3}}} x^{\frac{5}{3}} = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.8

Cancel the common factor of $x^{\frac{5}{3}}$ .

Tap for more steps...

Step 2.2.3.2.1.8.1

Cancel the common factor.

$4 x + \frac{2}{x^{\frac{5}{3}}} x^{\frac{5}{3}} = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.2.1.8.2

Rewrite the expression.

$4 x + 2 = 0 (9 x^{\frac{5}{3}})$

Step 2.2.3.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.3.1

Multiply $0 (9 x^{\frac{5}{3}})$ .

Tap for more steps...

Step 2.2.3.3.1.1

Multiply $9$ by $0$ .

$4 x + 2 = 0 x^{\frac{5}{3}}$

Step 2.2.3.3.1.2

Multiply $0$ by $x^{\frac{5}{3}}$ .

$4 x + 2 = 0$

Step 2.2.4

Solve the equation.

Tap for more steps...

Step 2.2.4.1

Subtract $2$ from both sides of the equation.

$4 x = - 2$

Step 2.2.4.2

Divide each term in $4 x = - 2$ by $4$ and simplify.

Tap for more steps...

Step 2.2.4.2.1

Divide each term in $4 x = - 2$ by $4$ .

$\frac{4 x}{4} = \frac{- 2}{4}$

Step 2.2.4.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.4.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.2.4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 2}{4}$

Step 2.2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{4}$

Step 2.2.4.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.4.2.3.1

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

Tap for more steps...

Step 2.2.4.2.3.1.1

Factor $2$ out of $- 2$ .

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

Step 2.2.4.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.2.4.2.3.1.2.1

Factor $2$ out of $4$ .

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

Step 2.2.4.2.3.1.2.2

Cancel the common factor.

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

Step 2.2.4.2.3.1.2.3

Rewrite the expression.

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

Step 2.2.4.2.3.2

Move the negative in front of the fraction.

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

Step 3

Find the domain of $f (x) = x^{\frac{4}{3}} - x^{\frac{1}{3}}$ .

Tap for more steps...

Step 3.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 3.1.1

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

$\sqrt[3]{x^{4}} - x^{\frac{1}{3}}$

Step 3.1.2

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

$\sqrt[3]{x^{4}} - \sqrt[3]{x^{1}}$

Step 3.1.3

Anything raised to $1$ is the base itself.

$\sqrt[3]{x^{4}} - \sqrt[3]{x}$

Step 3.2

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - \frac{1}{2})$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 5.1

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

$f'' (- 3) = \frac{4}{9 {(- 3)}^{\frac{2}{3}}} + \frac{2}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

To write $\frac{4}{9 {(- 3)}^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{{(- 3)}^{\frac{3}{3}}}{{(- 3)}^{\frac{3}{3}}}$ .

$f'' (- 3) = \frac{4}{9 {(- 3)}^{\frac{2}{3}}} \cdot \frac{{(- 3)}^{\frac{3}{3}}}{{(- 3)}^{\frac{3}{3}}} + \frac{2}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.2

Write each expression with a common denominator of $9 {(- 3)}^{\frac{5}{3}}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.2.2.1

Multiply $\frac{4}{9 {(- 3)}^{\frac{2}{3}}}$ by $\frac{{(- 3)}^{\frac{3}{3}}}{{(- 3)}^{\frac{3}{3}}}$ .

$f'' (- 3) = \frac{4 {(- 3)}^{\frac{3}{3}}}{9 {(- 3)}^{\frac{2}{3}} {(- 3)}^{\frac{3}{3}}} + \frac{2}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.2.2

Multiply ${(- 3)}^{\frac{2}{3}}$ by ${(- 3)}^{\frac{3}{3}}$ by adding the exponents.

Tap for more steps...

Step 5.2.2.2.1

Move ${(- 3)}^{\frac{3}{3}}$ .

$f'' (- 3) = \frac{4 {(- 3)}^{\frac{3}{3}}}{9 ({(- 3)}^{\frac{3}{3}} {(- 3)}^{\frac{2}{3}})} + \frac{2}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.2.2.2

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

$f'' (- 3) = \frac{4 {(- 3)}^{\frac{3}{3}}}{9 {(- 3)}^{\frac{3}{3} + \frac{2}{3}}} + \frac{2}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.2.2.3

Combine the numerators over the common denominator.

$f'' (- 3) = \frac{4 {(- 3)}^{\frac{3}{3}}}{9 {(- 3)}^{\frac{3 + 2}{3}}} + \frac{2}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.2.2.4

Add $3$ and $2$ .

$f'' (- 3) = \frac{4 {(- 3)}^{\frac{3}{3}}}{9 {(- 3)}^{\frac{5}{3}}} + \frac{2}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.3

Combine the numerators over the common denominator.

$f'' (- 3) = \frac{4 {(- 3)}^{\frac{3}{3}} + 2}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.4

Simplify the numerator.

Tap for more steps...

Step 5.2.4.1

Divide $3$ by $3$ .

$f'' (- 3) = \frac{4 (- 3) + 2}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.4.2

Raise $- 3$ to the power of $1$ .

$f'' (- 3) = \frac{4 \cdot - 3 + 2}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.4.3

Multiply $4$ by $- 3$ .

$f'' (- 3) = \frac{- 12 + 2}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.4.4

Add $- 12$ and $2$ .

$f'' (- 3) = \frac{- 10}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.5

Move the negative in front of the fraction.

$f'' (- 3) = - \frac{10}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.2.6

The final answer is $- \frac{10}{9 {(- 3)}^{\frac{5}{3}}}$ .

$- \frac{10}{9 {(- 3)}^{\frac{5}{3}}}$

Step 5.3

The graph is concave up on the interval $(- \infty, - \frac{1}{2})$ because $f'' (- 3)$ is positive.

The graph is concave up

Step 6

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

The graph is concave up

Step 7