Calculus Examples

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

Step 1

Find the second derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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 $g (x) = x^{3}$ .

Tap for more steps...

Step 1.1.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{3}$ .

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

Step 1.1.2.3.2

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

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

Step 1.1.2.3.3

Replace all occurrences of $u_{1}$ with $x^{3}$ .

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

Tap for more steps...

Step 1.1.2.5.1

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

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

Step 1.1.2.5.2

Multiply $3$ by $- 2$ .

$2 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [- \frac{3}{x^{2}}]$

Step 1.1.2.6

Multiply $3$ by $- 1$ .

$2 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [- \frac{3}{x^{2}}]$

Step 1.1.2.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.1.2.7.1

Move $x^{2}$ .

$2 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [- \frac{3}{x^{2}}]$

Step 1.1.2.7.2

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

$2 (- 3 x^{2 - 6}) + \frac{d}{d x} [- \frac{3}{x^{2}}]$

Step 1.1.2.7.3

Subtract $6$ from $2$ .

$2 (- 3 x^{- 4}) + \frac{d}{d x} [- \frac{3}{x^{2}}]$

Step 1.1.2.8

Multiply $- 3$ by $2$ .

$- 6 x^{- 4} + \frac{d}{d x} [- \frac{3}{x^{2}}]$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

$- 6 x^{- 4} - 3 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.1.3.2

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

$- 6 x^{- 4} - 3 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 1.1.3.3

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 $g (x) = x^{2}$ .

Tap for more steps...

Step 1.1.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{2}$ .

$- 6 x^{- 4} - 3 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{2}])$

Step 1.1.3.3.2

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

$- 6 x^{- 4} - 3 (- {u_{2}}^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.1.3.3.3

Replace all occurrences of $u_{2}$ with $x^{2}$ .

$- 6 x^{- 4} - 3 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.1.3.4

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

$- 6 x^{- 4} - 3 (- {(x^{2})}^{- 2} (2 x))$

Step 1.1.3.5

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

Tap for more steps...

Step 1.1.3.5.1

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

$- 6 x^{- 4} - 3 (- x^{2 \cdot - 2} (2 x))$

Step 1.1.3.5.2

Multiply $2$ by $- 2$ .

$- 6 x^{- 4} - 3 (- x^{- 4} (2 x))$

Step 1.1.3.6

Multiply $2$ by $- 1$ .

$- 6 x^{- 4} - 3 (- 2 x^{- 4} x)$

Step 1.1.3.7

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

$- 6 x^{- 4} - 3 (- 2 (x^{1} x^{- 4}))$

Step 1.1.3.8

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

$- 6 x^{- 4} - 3 (- 2 x^{1 - 4})$

Step 1.1.3.9

Subtract $4$ from $1$ .

$- 6 x^{- 4} - 3 (- 2 x^{- 3})$

Step 1.1.3.10

Multiply $- 2$ by $- 3$ .

$- 6 x^{- 4} + 6 x^{- 3}$

Step 1.1.4

Simplify.

Tap for more steps...

Step 1.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 6 \frac{1}{x^{4}} + 6 x^{- 3}$

Step 1.1.4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 6 \frac{1}{x^{4}} + 6 \frac{1}{x^{3}}$

Step 1.1.4.3

Combine terms.

Tap for more steps...

Step 1.1.4.3.1

Combine $- 6$ and $\frac{1}{x^{4}}$ .

$\frac{- 6}{x^{4}} + 6 \frac{1}{x^{3}}$

Step 1.1.4.3.2

Move the negative in front of the fraction.

$- \frac{6}{x^{4}} + 6 \frac{1}{x^{3}}$

Step 1.1.4.3.3

Combine $6$ and $\frac{1}{x^{3}}$ .

$- \frac{6}{x^{4}} + \frac{6}{x^{3}}$

Step 1.1.4.4

Reorder terms.

$f' (x) = \frac{6}{x^{3}} - \frac{6}{x^{4}}$

Step 1.2

Find the second derivative.

Tap for more steps...

Step 1.2.1

By the Sum Rule, the derivative of $\frac{6}{x^{3}} - \frac{6}{x^{4}}$ with respect to $x$ is $\frac{d}{d x} [\frac{6}{x^{3}}] + \frac{d}{d x} [- \frac{6}{x^{4}}]$ .

$\frac{d}{d x} [\frac{6}{x^{3}}] + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [\frac{6}{x^{3}}]$ .

Tap for more steps...

Step 1.2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6}{x^{3}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$6 \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.2

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

$6 \frac{d}{d x} [{(x^{3})}^{- 1}] + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.3

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 $g (x) = x^{3}$ .

Tap for more steps...

Step 1.2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{3}$ .

$6 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.3.2

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

$6 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.3.3

Replace all occurrences of $u_{1}$ with $x^{3}$ .

$6 (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.4

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

$6 (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.5

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

Tap for more steps...

Step 1.2.2.5.1

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

$6 (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.5.2

Multiply $3$ by $- 2$ .

$6 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.6

Multiply $3$ by $- 1$ .

$6 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.2.2.7.1

Move $x^{2}$ .

$6 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.7.2

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

$6 (- 3 x^{2 - 6}) + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.7.3

Subtract $6$ from $2$ .

$6 (- 3 x^{- 4}) + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.2.8

Multiply $- 3$ by $6$ .

$- 18 x^{- 4} + \frac{d}{d x} [- \frac{6}{x^{4}}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- \frac{6}{x^{4}}]$ .

Tap for more steps...

Step 1.2.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- \frac{6}{x^{4}}$ with respect to $x$ is $- 6 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$- 18 x^{- 4} - 6 \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 1.2.3.2

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$- 18 x^{- 4} - 6 \frac{d}{d x} [{(x^{4})}^{- 1}]$

Step 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 $g (x) = x^{4}$ .

Tap for more steps...

Step 1.2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{4}$ .

$- 18 x^{- 4} - 6 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{4}])$

Step 1.2.3.3.2

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

$- 18 x^{- 4} - 6 (- {u_{2}}^{- 2} \frac{d}{d x} [x^{4}])$

Step 1.2.3.3.3

Replace all occurrences of $u_{2}$ with $x^{4}$ .

$- 18 x^{- 4} - 6 (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}])$

Step 1.2.3.4

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

$- 18 x^{- 4} - 6 (- {(x^{4})}^{- 2} (4 x^{3}))$

Step 1.2.3.5

Multiply the exponents in ${(x^{4})}^{- 2}$ .

Tap for more steps...

Step 1.2.3.5.1

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

$- 18 x^{- 4} - 6 (- x^{4 \cdot - 2} (4 x^{3}))$

Step 1.2.3.5.2

Multiply $4$ by $- 2$ .

$- 18 x^{- 4} - 6 (- x^{- 8} (4 x^{3}))$

Step 1.2.3.6

Multiply $4$ by $- 1$ .

$- 18 x^{- 4} - 6 (- 4 x^{- 8} x^{3})$

Step 1.2.3.7

Multiply $x^{- 8}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 1.2.3.7.1

Move $x^{3}$ .

$- 18 x^{- 4} - 6 (- 4 (x^{3} x^{- 8}))$

Step 1.2.3.7.2

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

$- 18 x^{- 4} - 6 (- 4 x^{3 - 8})$

Step 1.2.3.7.3

Subtract $8$ from $3$ .

$- 18 x^{- 4} - 6 (- 4 x^{- 5})$

Step 1.2.3.8

Multiply $- 4$ by $- 6$ .

$- 18 x^{- 4} + 24 x^{- 5}$

Step 1.2.4

Simplify.

Tap for more steps...

Step 1.2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 18 \frac{1}{x^{4}} + 24 x^{- 5}$

Step 1.2.4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 18 \frac{1}{x^{4}} + 24 \frac{1}{x^{5}}$

Step 1.2.4.3

Combine terms.

Tap for more steps...

Step 1.2.4.3.1

Combine $- 18$ and $\frac{1}{x^{4}}$ .

$\frac{- 18}{x^{4}} + 24 \frac{1}{x^{5}}$

Step 1.2.4.3.2

Move the negative in front of the fraction.

$- \frac{18}{x^{4}} + 24 \frac{1}{x^{5}}$

Step 1.2.4.3.3

Combine $24$ and $\frac{1}{x^{5}}$ .

$f'' (x) = - \frac{18}{x^{4}} + \frac{24}{x^{5}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{18}{x^{4}} + \frac{24}{x^{5}}$ .

$- \frac{18}{x^{4}} + \frac{24}{x^{5}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $- \frac{18}{x^{4}} + \frac{24}{x^{5}} = 0$ .

Tap for more steps...

Step 2.1

Set the second derivative equal to $0$ .

$- \frac{18}{x^{4}} + \frac{24}{x^{5}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.2.1

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

$x^{4}, x^{5}, 1$

Step 2.2.2

Since $x^{4}, x^{5}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part $x^{4}, x^{5}$ .

Step 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 2.2.4

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

Not prime

Step 2.2.5

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

$1$

Step 2.2.6

The factors for $x^{4}$ are $x \cdot x \cdot x \cdot x$ , which is $x$ multiplied by each other $4$ times.

$x^{4} = x \cdot x \cdot x \cdot x$

$x$ occurs $4$ times.

Step 2.2.7

The factors for $x^{5}$ are $x \cdot x \cdot x \cdot x \cdot x$ , which is $x$ multiplied by each other $5$ times.

$x^{5} = x \cdot x \cdot x \cdot x \cdot x$

$x$ occurs $5$ times.

Step 2.2.8

The LCM of $x^{4}, x^{5}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x \cdot x \cdot x \cdot x$

Step 2.2.9

Simplify $x \cdot x \cdot x \cdot x \cdot x$ .

Tap for more steps...

Step 2.2.9.1

Multiply $x$ by $x$ .

$x^{2} \cdot x \cdot x \cdot x$

Step 2.2.9.2

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.2.9.2.1

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 2.2.9.2.1.1

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

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

Step 2.2.9.2.1.2

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

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

Step 2.2.9.2.2

Add $2$ and $1$ .

$x^{3} \cdot x \cdot x$

Step 2.2.9.3

Multiply $x^{3}$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.2.9.3.1

Multiply $x^{3}$ by $x$ .

Tap for more steps...

Step 2.2.9.3.1.1

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

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

Step 2.2.9.3.1.2

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

$x^{3 + 1} \cdot x$

Step 2.2.9.3.2

Add $3$ and $1$ .

$x^{4} \cdot x$

Step 2.2.9.4

Multiply $x^{4}$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.2.9.4.1

Multiply $x^{4}$ by $x$ .

Tap for more steps...

Step 2.2.9.4.1.1

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

$x^{4} \cdot x^{1}$

Step 2.2.9.4.1.2

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

$x^{4 + 1}$

Step 2.2.9.4.2

Add $4$ and $1$ .

$x^{5}$

Step 2.3

Multiply each term in $- \frac{18}{x^{4}} + \frac{24}{x^{5}} = 0$ by $x^{5}$ to eliminate the fractions.

Tap for more steps...

Step 2.3.1

Multiply each term in $- \frac{18}{x^{4}} + \frac{24}{x^{5}} = 0$ by $x^{5}$ .

$- \frac{18}{x^{4}} x^{5} + \frac{24}{x^{5}} x^{5} = 0 x^{5}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Simplify each term.

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor of $x^{4}$ .

Tap for more steps...

Step 2.3.2.1.1.1

Move the leading negative in $- \frac{18}{x^{4}}$ into the numerator.

$\frac{- 18}{x^{4}} x^{5} + \frac{24}{x^{5}} x^{5} = 0 x^{5}$

Step 2.3.2.1.1.2

Factor $x^{4}$ out of $x^{5}$ .

$\frac{- 18}{x^{4}} (x^{4} x) + \frac{24}{x^{5}} x^{5} = 0 x^{5}$

Step 2.3.2.1.1.3

Cancel the common factor.

$\frac{- 18}{x^{4}} (x^{4} x) + \frac{24}{x^{5}} x^{5} = 0 x^{5}$

Step 2.3.2.1.1.4

Rewrite the expression.

$- 18 x + \frac{24}{x^{5}} x^{5} = 0 x^{5}$

Step 2.3.2.1.2

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

Tap for more steps...

Step 2.3.2.1.2.1

Cancel the common factor.

$- 18 x + \frac{24}{x^{5}} x^{5} = 0 x^{5}$

Step 2.3.2.1.2.2

Rewrite the expression.

$- 18 x + 24 = 0 x^{5}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Multiply $0$ by $x^{5}$ .

$- 18 x + 24 = 0$

Step 2.4

Solve the equation.

Tap for more steps...

Step 2.4.1

Subtract $24$ from both sides of the equation.

$- 18 x = - 24$

Step 2.4.2

Divide each term in $- 18 x = - 24$ by $- 18$ and simplify.

Tap for more steps...

Step 2.4.2.1

Divide each term in $- 18 x = - 24$ by $- 18$ .

$\frac{- 18 x}{- 18} = \frac{- 24}{- 18}$

Step 2.4.2.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.2.1

Cancel the common factor of $- 18$ .

Tap for more steps...

Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{- 18 x}{- 18} = \frac{- 24}{- 18}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 24}{- 18}$

Step 2.4.2.3

Simplify the right side.

Tap for more steps...

Step 2.4.2.3.1

Cancel the common factor of $- 24$ and $- 18$ .

Tap for more steps...

Step 2.4.2.3.1.1

Factor $- 6$ out of $- 24$ .

$x = \frac{- 6 (4)}{- 18}$

Step 2.4.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.4.2.3.1.2.1

Factor $- 6$ out of $- 18$ .

$x = \frac{- 6 \cdot 4}{- 6 \cdot 3}$

Step 2.4.2.3.1.2.2

Cancel the common factor.

$x = \frac{- 6 \cdot 4}{- 6 \cdot 3}$

Step 2.4.2.3.1.2.3

Rewrite the expression.

$x = \frac{4}{3}$

Step 3

Find the points where the second derivative is $0$ .

Tap for more steps...

Step 3.1

Substitute $\frac{4}{3}$ in $f (x) = \frac{2}{x^{3}} - \frac{3}{x^{2}}$ to find the value of $y$ .

Tap for more steps...

Step 3.1.1

Replace the variable $x$ with $\frac{4}{3}$ in the expression.

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

Step 3.1.2

Simplify the result.

Tap for more steps...

Step 3.1.2.1

Simplify each term.

Tap for more steps...

Step 3.1.2.1.1

Simplify the denominator.

Tap for more steps...

Step 3.1.2.1.1.1

Apply the product rule to $\frac{4}{3}$ .

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

Step 3.1.2.1.1.2

Raise $4$ to the power of $3$ .

$f (\frac{4}{3}) = \frac{2}{\frac{64}{3^{3}}} - \frac{3}{{(\frac{4}{3})}^{2}}$

Step 3.1.2.1.1.3

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

$f (\frac{4}{3}) = \frac{2}{\frac{64}{27}} - \frac{3}{{(\frac{4}{3})}^{2}}$

Step 3.1.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{4}{3}) = 2 (\frac{27}{64}) - \frac{3}{{(\frac{4}{3})}^{2}}$

Step 3.1.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.2.1.3.1

Factor $2$ out of $64$ .

$f (\frac{4}{3}) = 2 (\frac{27}{2 (32)}) - \frac{3}{{(\frac{4}{3})}^{2}}$

Step 3.1.2.1.3.2

Cancel the common factor.

$f (\frac{4}{3}) = 2 (\frac{27}{2 \cdot 32}) - \frac{3}{{(\frac{4}{3})}^{2}}$

Step 3.1.2.1.3.3

Rewrite the expression.

$f (\frac{4}{3}) = \frac{27}{32} - \frac{3}{{(\frac{4}{3})}^{2}}$

Step 3.1.2.1.4

Simplify the denominator.

Tap for more steps...

Step 3.1.2.1.4.1

Apply the product rule to $\frac{4}{3}$ .

$f (\frac{4}{3}) = \frac{27}{32} - \frac{3}{\frac{4^{2}}{3^{2}}}$

Step 3.1.2.1.4.2

Raise $4$ to the power of $2$ .

$f (\frac{4}{3}) = \frac{27}{32} - \frac{3}{\frac{16}{3^{2}}}$

Step 3.1.2.1.4.3

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

$f (\frac{4}{3}) = \frac{27}{32} - \frac{3}{\frac{16}{9}}$

Step 3.1.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{4}{3}) = \frac{27}{32} - (3 (\frac{9}{16}))$

Step 3.1.2.1.6

Multiply $3 (\frac{9}{16})$ .

Tap for more steps...

Step 3.1.2.1.6.1

Combine $3$ and $\frac{9}{16}$ .

$f (\frac{4}{3}) = \frac{27}{32} - \frac{3 \cdot 9}{16}$

Step 3.1.2.1.6.2

Multiply $3$ by $9$ .

$f (\frac{4}{3}) = \frac{27}{32} - \frac{27}{16}$

Step 3.1.2.2

To write $- \frac{27}{16}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{4}{3}) = \frac{27}{32} - \frac{27}{16} \cdot \frac{2}{2}$

Step 3.1.2.3

Write each expression with a common denominator of $32$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1.2.3.1

Multiply $\frac{27}{16}$ by $\frac{2}{2}$ .

$f (\frac{4}{3}) = \frac{27}{32} - \frac{27 \cdot 2}{16 \cdot 2}$

Step 3.1.2.3.2

Multiply $16$ by $2$ .

$f (\frac{4}{3}) = \frac{27}{32} - \frac{27 \cdot 2}{32}$

Step 3.1.2.4

Combine the numerators over the common denominator.

$f (\frac{4}{3}) = \frac{27 - 27 \cdot 2}{32}$

Step 3.1.2.5

Simplify the numerator.

Tap for more steps...

Step 3.1.2.5.1

Multiply $- 27$ by $2$ .

$f (\frac{4}{3}) = \frac{27 - 54}{32}$

Step 3.1.2.5.2

Subtract $54$ from $27$ .

$f (\frac{4}{3}) = \frac{- 27}{32}$

Step 3.1.2.6

Move the negative in front of the fraction.

$f (\frac{4}{3}) = - \frac{27}{32}$

Step 3.1.2.7

The final answer is $- \frac{27}{32}$ .

$- \frac{27}{32}$

Step 3.2

The point found by substituting $\frac{4}{3}$ in $f (x) = \frac{2}{x^{3}} - \frac{3}{x^{2}}$ is $(\frac{4}{3}, - \frac{27}{32})$ . This point can be an inflection point.

$(\frac{4}{3}, - \frac{27}{32})$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, \frac{4}{3}) \cup (\frac{4}{3}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{4}{3})$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $1.2\overline{3}$ in the expression.

$f'' (1.2\overline{3}) = - \frac{18}{{(1.2\overline{3})}^{4}} + \frac{24}{{(1.2\overline{3})}^{5}}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Raise $1.2\overline{3}$ to the power of $4$ .

$f'' (1.2\overline{3}) = - \frac{18}{2.31377901} + \frac{24}{{(1.2\overline{3})}^{5}}$

Step 5.2.1.2

Divide $18$ by $2.31377901$ .

$f'' (1.2\overline{3}) = - 1 \cdot 7.77948105 + \frac{24}{{(1.2\overline{3})}^{5}}$

Step 5.2.1.3

Multiply $- 1$ by $7.77948105$ .

$f'' (1.2\overline{3}) = - 7.77948105 + \frac{24}{{(1.2\overline{3})}^{5}}$

Step 5.2.1.4

Raise $1.2\overline{3}$ to the power of $5$ .

$f'' (1.2\overline{3}) = - 7.77948105 + \frac{24}{2.85366078}$

Step 5.2.1.5

Divide $24$ by $2.85366078$ .

$f'' (1.2\overline{3}) = - 7.77948105 + 8.41024979$

Step 5.2.2

Add $- 7.77948105$ and $8.41024979$ .

$f'' (1.2\overline{3}) = 0.63076873$

Step 5.2.3

The final answer is $0.63076873$ .

$0.63076873$

Step 5.3

At $1.2\overline{3}$ , the second derivative is $0.63076873$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, \frac{4}{3})$ .

Increasing on $(- \infty, \frac{4}{3})$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(\frac{4}{3}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $1.4\overline{3}$ in the expression.

$f'' (1.4\overline{3}) = - \frac{18}{{(1.4\overline{3})}^{4}} + \frac{24}{{(1.4\overline{3})}^{5}}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Raise $1.4\overline{3}$ to the power of $4$ .

$f'' (1.4\overline{3}) = - \frac{18}{4.22074197} + \frac{24}{{(1.4\overline{3})}^{5}}$

Step 6.2.1.2

Divide $18$ by $4.22074197$ .

$f'' (1.4\overline{3}) = - 1 \cdot 4.26465301 + \frac{24}{{(1.4\overline{3})}^{5}}$

Step 6.2.1.3

Multiply $- 1$ by $4.26465301$ .

$f'' (1.4\overline{3}) = - 4.26465301 + \frac{24}{{(1.4\overline{3})}^{5}}$

Step 6.2.1.4

Raise $1.4\overline{3}$ to the power of $5$ .

$f'' (1.4\overline{3}) = - 4.26465301 + \frac{24}{6.04973016}$

Step 6.2.1.5

Divide $24$ by $6.04973016$ .

$f'' (1.4\overline{3}) = - 4.26465301 + 3.96711908$

Step 6.2.2

Add $- 4.26465301$ and $3.96711908$ .

$f'' (1.4\overline{3}) = - 0.29753393$

Step 6.2.3

The final answer is $- 0.29753393$ .

$- 0.29753393$

Step 6.3

At $1.4\overline{3}$ , the second derivative is $- 0.29753393$ . Since this is negative, the second derivative is decreasing on the interval $(\frac{4}{3}, \infty)$

Decreasing on $(\frac{4}{3}, \infty)$ since $f'' (x) < 0$

Step 7

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(\frac{4}{3}, - \frac{27}{32})$ .

$(\frac{4}{3}, - \frac{27}{32})$

Step 8