Calculus Examples

$\frac{1}{5} x^{5} - x^{4} - 12 x^{3}$

Step 1

Write $\frac{1}{5} x^{5} - x^{4} - 12 x^{3}$ as a function.

$f (x) = \frac{1}{5} x^{5} - x^{4} - 12 x^{3}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $\frac{1}{5} x^{5} - x^{4} - 12 x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{5} x^{5}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [- 12 x^{3}]$ .

$\frac{d}{d x} [\frac{1}{5} x^{5}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2

Evaluate $\frac{d}{d x} [\frac{1}{5} x^{5}]$ .

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Step 2.1.2.1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{1}{5} x^{5}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [x^{5}]$ .

$\frac{1}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{1}{5} (5 x^{4}) + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2.3

Combine $5$ and $\frac{1}{5}$ .

$\frac{5}{5} x^{4} + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2.4

Combine $\frac{5}{5}$ and $x^{4}$ .

$\frac{5 x^{4}}{5} + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2.5

Cancel the common factor of $5$ .

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Step 2.1.2.5.1

Cancel the common factor.

$\frac{5 x^{4}}{5} + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2.5.2

Divide $x^{4}$ by $1$ .

$x^{4} + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.3

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

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Step 2.1.3.1

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

$x^{4} - \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.3.2

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

$x^{4} - (4 x^{3}) + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.3.3

Multiply $4$ by $- 1$ .

$x^{4} - 4 x^{3} + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.4

Evaluate $\frac{d}{d x} [- 12 x^{3}]$ .

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Step 2.1.4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{3}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{3}]$ .

$x^{4} - 4 x^{3} - 12 \frac{d}{d x} [x^{3}]$

Step 2.1.4.2

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

$x^{4} - 4 x^{3} - 12 (3 x^{2})$

Step 2.1.4.3

Multiply $3$ by $- 12$ .

$f' (x) = x^{4} - 4 x^{3} - 36 x^{2}$

Step 2.2

Find the second derivative.

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Step 2.2.1

Differentiate.

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Step 2.2.1.1

By the Sum Rule, the derivative of $x^{4} - 4 x^{3} - 36 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.1.2

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

$4 x^{3} + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.2

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

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Step 2.2.2.1

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

$4 x^{3} - 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.2.2

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

$4 x^{3} - 4 (3 x^{2}) + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.2.3

Multiply $3$ by $- 4$ .

$4 x^{3} - 12 x^{2} + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.3

Evaluate $\frac{d}{d x} [- 36 x^{2}]$ .

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Step 2.2.3.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 x^{2}$ with respect to $x$ is $- 36 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 12 x^{2} - 36 \frac{d}{d x} [x^{2}]$

Step 2.2.3.2

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

$4 x^{3} - 12 x^{2} - 36 (2 x)$

Step 2.2.3.3

Multiply $2$ by $- 36$ .

$f'' (x) = 4 x^{3} - 12 x^{2} - 72 x$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 12 x^{2} - 72 x$ .

$4 x^{3} - 12 x^{2} - 72 x$

Step 3

Set the second derivative equal to $0$ then solve the equation $4 x^{3} - 12 x^{2} - 72 x = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$4 x^{3} - 12 x^{2} - 72 x = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $4 x$ out of $4 x^{3} - 12 x^{2} - 72 x$ .

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Step 3.2.1.1

Factor $4 x$ out of $4 x^{3}$ .

$4 x (x^{2}) - 12 x^{2} - 72 x = 0$

Step 3.2.1.2

Factor $4 x$ out of $- 12 x^{2}$ .

$4 x (x^{2}) + 4 x (- 3 x) - 72 x = 0$

Step 3.2.1.3

Factor $4 x$ out of $- 72 x$ .

$4 x (x^{2}) + 4 x (- 3 x) + 4 x (- 18) = 0$

Step 3.2.1.4

Factor $4 x$ out of $4 x (x^{2}) + 4 x (- 3 x)$ .

$4 x (x^{2} - 3 x) + 4 x (- 18) = 0$

Step 3.2.1.5

Factor $4 x$ out of $4 x (x^{2} - 3 x) + 4 x (- 18)$ .

$4 x (x^{2} - 3 x - 18) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor $x^{2} - 3 x - 18$ using the AC method.

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Step 3.2.2.1.1

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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 3.2.2.1.2

Write the factored form using these integers.

$4 x ((x - 6) (x + 3)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$4 x (x - 6) (x + 3) = 0$

Step 3.3

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

$x = 0$

$x - 6 = 0$

$x + 3 = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.5.2

Add $6$ to both sides of the equation.

$x = 6$

Step 3.6

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.7

The final solution is all the values that make $4 x (x - 6) (x + 3) = 0$ true.

$x = 0, 6, - 3$

Step 4

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

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Step 4.1

Substitute $0$ in $f (x) = \frac{1}{5} x^{5} - x^{4} - 12 x^{3}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{1}{5} \cdot {(0)}^{5} - {(0)}^{4} - 12 {(0)}^{3}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$f (0) = \frac{1}{5} \cdot 0 - {(0)}^{4} - 12 {(0)}^{3}$

Step 4.1.2.1.2

Multiply $\frac{1}{5}$ by $0$ .

$f (0) = 0 - {(0)}^{4} - 12 {(0)}^{3}$

Step 4.1.2.1.3

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

$f (0) = 0 - 0 - 12 {(0)}^{3}$

Step 4.1.2.1.4

Multiply $- 1$ by $0$ .

$f (0) = 0 + 0 - 12 {(0)}^{3}$

Step 4.1.2.1.5

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

$f (0) = 0 + 0 - 12 \cdot 0$

Step 4.1.2.1.6

Multiply $- 12$ by $0$ .

$f (0) = 0 + 0 + 0$

Step 4.1.2.2

Simplify by adding numbers.

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Step 4.1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 4.1.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 4.1.2.3

The final answer is $0$ .

$0$

Step 4.2

The point found by substituting $0$ in $f (x) = \frac{1}{5} x^{5} - x^{4} - 12 x^{3}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4.3

Substitute $6$ in $f (x) = \frac{1}{5} x^{5} - x^{4} - 12 x^{3}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{1}{5} \cdot {(6)}^{5} - {(6)}^{4} - 12 {(6)}^{3}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Raise $6$ to the power of $5$ .

$f (6) = \frac{1}{5} \cdot 7776 - {(6)}^{4} - 12 {(6)}^{3}$

Step 4.3.2.1.2

Combine $\frac{1}{5}$ and $7776$ .

$f (6) = \frac{7776}{5} - {(6)}^{4} - 12 {(6)}^{3}$

Step 4.3.2.1.3

Raise $6$ to the power of $4$ .

$f (6) = \frac{7776}{5} - 1 \cdot 1296 - 12 {(6)}^{3}$

Step 4.3.2.1.4

Multiply $- 1$ by $1296$ .

$f (6) = \frac{7776}{5} - 1296 - 12 {(6)}^{3}$

Step 4.3.2.1.5

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

$f (6) = \frac{7776}{5} - 1296 - 12 \cdot 216$

Step 4.3.2.1.6

Multiply $- 12$ by $216$ .

$f (6) = \frac{7776}{5} - 1296 - 2592$

Step 4.3.2.2

Find the common denominator.

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Step 4.3.2.2.1

Write $- 1296$ as a fraction with denominator $1$ .

$f (6) = \frac{7776}{5} + \frac{- 1296}{1} - 2592$

Step 4.3.2.2.2

Multiply $\frac{- 1296}{1}$ by $\frac{5}{5}$ .

$f (6) = \frac{7776}{5} + \frac{- 1296}{1} \cdot \frac{5}{5} - 2592$

Step 4.3.2.2.3

Multiply $\frac{- 1296}{1}$ by $\frac{5}{5}$ .

$f (6) = \frac{7776}{5} + \frac{- 1296 \cdot 5}{5} - 2592$

Step 4.3.2.2.4

Write $- 2592$ as a fraction with denominator $1$ .

$f (6) = \frac{7776}{5} + \frac{- 1296 \cdot 5}{5} + \frac{- 2592}{1}$

Step 4.3.2.2.5

Multiply $\frac{- 2592}{1}$ by $\frac{5}{5}$ .

$f (6) = \frac{7776}{5} + \frac{- 1296 \cdot 5}{5} + \frac{- 2592}{1} \cdot \frac{5}{5}$

Step 4.3.2.2.6

Multiply $\frac{- 2592}{1}$ by $\frac{5}{5}$ .

$f (6) = \frac{7776}{5} + \frac{- 1296 \cdot 5}{5} + \frac{- 2592 \cdot 5}{5}$

Step 4.3.2.3

Combine the numerators over the common denominator.

$f (6) = \frac{7776 - 1296 \cdot 5 - 2592 \cdot 5}{5}$

Step 4.3.2.4

Simplify each term.

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Step 4.3.2.4.1

Multiply $- 1296$ by $5$ .

$f (6) = \frac{7776 - 6480 - 2592 \cdot 5}{5}$

Step 4.3.2.4.2

Multiply $- 2592$ by $5$ .

$f (6) = \frac{7776 - 6480 - 12960}{5}$

Step 4.3.2.5

Simplify the expression.

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Step 4.3.2.5.1

Subtract $6480$ from $7776$ .

$f (6) = \frac{1296 - 12960}{5}$

Step 4.3.2.5.2

Subtract $12960$ from $1296$ .

$f (6) = \frac{- 11664}{5}$

Step 4.3.2.5.3

Move the negative in front of the fraction.

$f (6) = - \frac{11664}{5}$

Step 4.3.2.6

The final answer is $- \frac{11664}{5}$ .

$- \frac{11664}{5}$

Step 4.4

The point found by substituting $6$ in $f (x) = \frac{1}{5} x^{5} - x^{4} - 12 x^{3}$ is $(6, - \frac{11664}{5})$ . This point can be an inflection point.

$(6, - \frac{11664}{5})$

Step 4.5

Substitute $- 3$ in $f (x) = \frac{1}{5} x^{5} - x^{4} - 12 x^{3}$ to find the value of $y$ .

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Step 4.5.1

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

$f (- 3) = \frac{1}{5} \cdot {(- 3)}^{5} - {(- 3)}^{4} - 12 {(- 3)}^{3}$

Step 4.5.2

Simplify the result.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

Raise $- 3$ to the power of $5$ .

$f (- 3) = \frac{1}{5} \cdot - 243 - {(- 3)}^{4} - 12 {(- 3)}^{3}$

Step 4.5.2.1.2

Combine $\frac{1}{5}$ and $- 243$ .

$f (- 3) = \frac{- 243}{5} - {(- 3)}^{4} - 12 {(- 3)}^{3}$

Step 4.5.2.1.3

Move the negative in front of the fraction.

$f (- 3) = - \frac{243}{5} - {(- 3)}^{4} - 12 {(- 3)}^{3}$

Step 4.5.2.1.4

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

$f (- 3) = - \frac{243}{5} - 1 \cdot 81 - 12 {(- 3)}^{3}$

Step 4.5.2.1.5

Multiply $- 1$ by $81$ .

$f (- 3) = - \frac{243}{5} - 81 - 12 {(- 3)}^{3}$

Step 4.5.2.1.6

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

$f (- 3) = - \frac{243}{5} - 81 - 12 \cdot - 27$

Step 4.5.2.1.7

Multiply $- 12$ by $- 27$ .

$f (- 3) = - \frac{243}{5} - 81 + 324$

Step 4.5.2.2

Find the common denominator.

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Step 4.5.2.2.1

Write $- 81$ as a fraction with denominator $1$ .

$f (- 3) = - \frac{243}{5} + \frac{- 81}{1} + 324$

Step 4.5.2.2.2

Multiply $\frac{- 81}{1}$ by $\frac{5}{5}$ .

$f (- 3) = - \frac{243}{5} + \frac{- 81}{1} \cdot \frac{5}{5} + 324$

Step 4.5.2.2.3

Multiply $\frac{- 81}{1}$ by $\frac{5}{5}$ .

$f (- 3) = - \frac{243}{5} + \frac{- 81 \cdot 5}{5} + 324$

Step 4.5.2.2.4

Write $324$ as a fraction with denominator $1$ .

$f (- 3) = - \frac{243}{5} + \frac{- 81 \cdot 5}{5} + \frac{324}{1}$

Step 4.5.2.2.5

Multiply $\frac{324}{1}$ by $\frac{5}{5}$ .

$f (- 3) = - \frac{243}{5} + \frac{- 81 \cdot 5}{5} + \frac{324}{1} \cdot \frac{5}{5}$

Step 4.5.2.2.6

Multiply $\frac{324}{1}$ by $\frac{5}{5}$ .

$f (- 3) = - \frac{243}{5} + \frac{- 81 \cdot 5}{5} + \frac{324 \cdot 5}{5}$

Step 4.5.2.3

Combine the numerators over the common denominator.

$f (- 3) = \frac{- 243 - 81 \cdot 5 + 324 \cdot 5}{5}$

Step 4.5.2.4

Simplify each term.

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Step 4.5.2.4.1

Multiply $- 81$ by $5$ .

$f (- 3) = \frac{- 243 - 405 + 324 \cdot 5}{5}$

Step 4.5.2.4.2

Multiply $324$ by $5$ .

$f (- 3) = \frac{- 243 - 405 + 1620}{5}$

Step 4.5.2.5

Simplify by adding and subtracting.

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Step 4.5.2.5.1

Subtract $405$ from $- 243$ .

$f (- 3) = \frac{- 648 + 1620}{5}$

Step 4.5.2.5.2

Add $- 648$ and $1620$ .

$f (- 3) = \frac{972}{5}$

Step 4.5.2.6

The final answer is $\frac{972}{5}$ .

$\frac{972}{5}$

Step 4.6

The point found by substituting $- 3$ in $f (x) = \frac{1}{5} x^{5} - x^{4} - 12 x^{3}$ is $(- 3, \frac{972}{5})$ . This point can be an inflection point.

$(- 3, \frac{972}{5})$

Step 4.7

Determine the points that could be inflection points.

$(0, 0), (6, - \frac{11664}{5}), (- 3, \frac{972}{5})$

Step 5

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

$(- \infty, - 3) \cup (- 3, 0) \cup (0, 6) \cup (6, \infty)$

Step 6

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

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Step 6.1

Replace the variable $x$ with $- 3.1$ in the expression.

$f'' (- 3.1) = 4 {(- 3.1)}^{3} - 12 {(- 3.1)}^{2} - 72 \cdot - 3.1$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 3.1$ to the power of $3$ .

$f'' (- 3.1) = 4 \cdot - 29.791 - 12 {(- 3.1)}^{2} - 72 \cdot - 3.1$

Step 6.2.1.2

Multiply $4$ by $- 29.791$ .

$f'' (- 3.1) = - 119.164 - 12 {(- 3.1)}^{2} - 72 \cdot - 3.1$

Step 6.2.1.3

Raise $- 3.1$ to the power of $2$ .

$f'' (- 3.1) = - 119.164 - 12 \cdot 9.61 - 72 \cdot - 3.1$

Step 6.2.1.4

Multiply $- 12$ by $9.61$ .

$f'' (- 3.1) = - 119.164 - 115.32 - 72 \cdot - 3.1$

Step 6.2.1.5

Multiply $- 72$ by $- 3.1$ .

$f'' (- 3.1) = - 119.164 - 115.32 + 223.2$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $115.32$ from $- 119.164$ .

$f'' (- 3.1) = - 234.484 + 223.2$

Step 6.2.2.2

Add $- 234.484$ and $223.2$ .

$f'' (- 3.1) = - 11.284$

Step 6.2.3

The final answer is $- 11.284$ .

$- 11.284$

Step 6.3

At $- 3.1$ , the second derivative is $- 11.284$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 3)$

Decreasing on $(- \infty, - 3)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 3, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

$f'' (- \frac{3}{2}) = 4 {(- \frac{3}{2})}^{3} - 12 {(- \frac{3}{2})}^{2} - 72 (- \frac{3}{2})$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.2.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$f'' (- \frac{3}{2}) = 4 ({(- 1)}^{3} {(\frac{3}{2})}^{3}) - 12 {(- \frac{3}{2})}^{2} - 72 (- \frac{3}{2})$

Step 7.2.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$f'' (- \frac{3}{2}) = 4 ({(- 1)}^{3} (\frac{3^{3}}{2^{3}})) - 12 {(- \frac{3}{2})}^{2} - 72 (- \frac{3}{2})$

Step 7.2.1.2

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

$f'' (- \frac{3}{2}) = 4 (- \frac{3^{3}}{2^{3}}) - 12 {(- \frac{3}{2})}^{2} - 72 (- \frac{3}{2})$

Step 7.2.1.3

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

$f'' (- \frac{3}{2}) = 4 (- \frac{27}{2^{3}}) - 12 {(- \frac{3}{2})}^{2} - 72 (- \frac{3}{2})$

Step 7.2.1.4

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

$f'' (- \frac{3}{2}) = 4 (- \frac{27}{8}) - 12 {(- \frac{3}{2})}^{2} - 72 (- \frac{3}{2})$

Step 7.2.1.5

Cancel the common factor of $4$ .

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Step 7.2.1.5.1

Move the leading negative in $- \frac{27}{8}$ into the numerator.

$f'' (- \frac{3}{2}) = 4 (\frac{- 27}{8}) - 12 {(- \frac{3}{2})}^{2} - 72 (- \frac{3}{2})$

Step 7.2.1.5.2

Factor $4$ out of $8$ .

$f'' (- \frac{3}{2}) = 4 (\frac{- 27}{4 (2)}) - 12 {(- \frac{3}{2})}^{2} - 72 (- \frac{3}{2})$

Step 7.2.1.5.3

Cancel the common factor.

$f'' (- \frac{3}{2}) = 4 (\frac{- 27}{4 \cdot 2}) - 12 {(- \frac{3}{2})}^{2} - 72 (- \frac{3}{2})$

Step 7.2.1.5.4

Rewrite the expression.

$f'' (- \frac{3}{2}) = \frac{- 27}{2} - 12 {(- \frac{3}{2})}^{2} - 72 (- \frac{3}{2})$

Step 7.2.1.6

Move the negative in front of the fraction.

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 12 {(- \frac{3}{2})}^{2} - 72 (- \frac{3}{2})$

Step 7.2.1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.2.1.7.1

Apply the product rule to $- \frac{3}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 12 ({(- 1)}^{2} {(\frac{3}{2})}^{2}) - 72 (- \frac{3}{2})$

Step 7.2.1.7.2

Apply the product rule to $\frac{3}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 12 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}})) - 72 (- \frac{3}{2})$

Step 7.2.1.8

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

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 12 (1 (\frac{3^{2}}{2^{2}})) - 72 (- \frac{3}{2})$

Step 7.2.1.9

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

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 12 \frac{3^{2}}{2^{2}} - 72 (- \frac{3}{2})$

Step 7.2.1.10

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

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 12 \frac{9}{2^{2}} - 72 (- \frac{3}{2})$

Step 7.2.1.11

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

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 12 (\frac{9}{4}) - 72 (- \frac{3}{2})$

Step 7.2.1.12

Cancel the common factor of $4$ .

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Step 7.2.1.12.1

Factor $4$ out of $- 12$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} + 4 (- 3) (\frac{9}{4}) - 72 (- \frac{3}{2})$

Step 7.2.1.12.2

Cancel the common factor.

$f'' (- \frac{3}{2}) = - \frac{27}{2} + 4 \cdot (- 3 (\frac{9}{4})) - 72 (- \frac{3}{2})$

Step 7.2.1.12.3

Rewrite the expression.

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 3 \cdot 9 - 72 (- \frac{3}{2})$

Step 7.2.1.13

Multiply $- 3$ by $9$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 27 - 72 (- \frac{3}{2})$

Step 7.2.1.14

Cancel the common factor of $2$ .

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Step 7.2.1.14.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 27 - 72 (\frac{- 3}{2})$

Step 7.2.1.14.2

Factor $2$ out of $- 72$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 27 + 2 (- 36) (\frac{- 3}{2})$

Step 7.2.1.14.3

Cancel the common factor.

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 27 + 2 \cdot (- 36 (\frac{- 3}{2}))$

Step 7.2.1.14.4

Rewrite the expression.

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 27 - 36 \cdot - 3$

Step 7.2.1.15

Multiply $- 36$ by $- 3$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} - 27 + 108$

Step 7.2.2

Find the common denominator.

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Step 7.2.2.1

Write $- 27$ as a fraction with denominator $1$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} + \frac{- 27}{1} + 108$

Step 7.2.2.2

Multiply $\frac{- 27}{1}$ by $\frac{2}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} + \frac{- 27}{1} \cdot \frac{2}{2} + 108$

Step 7.2.2.3

Multiply $\frac{- 27}{1}$ by $\frac{2}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} + \frac{- 27 \cdot 2}{2} + 108$

Step 7.2.2.4

Write $108$ as a fraction with denominator $1$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} + \frac{- 27 \cdot 2}{2} + \frac{108}{1}$

Step 7.2.2.5

Multiply $\frac{108}{1}$ by $\frac{2}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} + \frac{- 27 \cdot 2}{2} + \frac{108}{1} \cdot \frac{2}{2}$

Step 7.2.2.6

Multiply $\frac{108}{1}$ by $\frac{2}{2}$ .

$f'' (- \frac{3}{2}) = - \frac{27}{2} + \frac{- 27 \cdot 2}{2} + \frac{108 \cdot 2}{2}$

Step 7.2.3

Combine the numerators over the common denominator.

$f'' (- \frac{3}{2}) = \frac{- 27 - 27 \cdot 2 + 108 \cdot 2}{2}$

Step 7.2.4

Simplify each term.

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Step 7.2.4.1

Multiply $- 27$ by $2$ .

$f'' (- \frac{3}{2}) = \frac{- 27 - 54 + 108 \cdot 2}{2}$

Step 7.2.4.2

Multiply $108$ by $2$ .

$f'' (- \frac{3}{2}) = \frac{- 27 - 54 + 216}{2}$

Step 7.2.5

Simplify by adding and subtracting.

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Step 7.2.5.1

Subtract $54$ from $- 27$ .

$f'' (- \frac{3}{2}) = \frac{- 81 + 216}{2}$

Step 7.2.5.2

Add $- 81$ and $216$ .

$f'' (- \frac{3}{2}) = \frac{135}{2}$

Step 7.2.6

The final answer is $\frac{135}{2}$ .

$\frac{135}{2}$

Step 7.3

At $- \frac{3}{2}$ , the second derivative is $67.5$ . Since this is positive, the second derivative is increasing on the interval $(- 3, 0)$ .

Increasing on $(- 3, 0)$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(0, 6)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

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

$f'' (3) = 4 {(3)}^{3} - 12 {(3)}^{2} - 72 \cdot 3$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

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

$f'' (3) = 4 \cdot 27 - 12 {(3)}^{2} - 72 \cdot 3$

Step 8.2.1.2

Multiply $4$ by $27$ .

$f'' (3) = 108 - 12 {(3)}^{2} - 72 \cdot 3$

Step 8.2.1.3

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

$f'' (3) = 108 - 12 \cdot 9 - 72 \cdot 3$

Step 8.2.1.4

Multiply $- 12$ by $9$ .

$f'' (3) = 108 - 108 - 72 \cdot 3$

Step 8.2.1.5

Multiply $- 72$ by $3$ .

$f'' (3) = 108 - 108 - 216$

Step 8.2.2

Simplify by subtracting numbers.

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Step 8.2.2.1

Subtract $108$ from $108$ .

$f'' (3) = 0 - 216$

Step 8.2.2.2

Subtract $216$ from $0$ .

$f'' (3) = - 216$

Step 8.2.3

The final answer is $- 216$ .

$- 216$

Step 8.3

At $3$ , the second derivative is $- 216$ . Since this is negative, the second derivative is decreasing on the interval $(0, 6)$

Decreasing on $(0, 6)$ since $f'' (x) < 0$

Step 9

Substitute a value from the interval $(6, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $6.1$ in the expression.

$f'' (6.1) = 4 {(6.1)}^{3} - 12 {(6.1)}^{2} - 72 \cdot 6.1$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Raise $6.1$ to the power of $3$ .

$f'' (6.1) = 4 \cdot 226.981 - 12 {(6.1)}^{2} - 72 \cdot 6.1$

Step 9.2.1.2

Multiply $4$ by $226.981$ .

$f'' (6.1) = 907.924 - 12 {(6.1)}^{2} - 72 \cdot 6.1$

Step 9.2.1.3

Raise $6.1$ to the power of $2$ .

$f'' (6.1) = 907.924 - 12 \cdot 37.21 - 72 \cdot 6.1$

Step 9.2.1.4

Multiply $- 12$ by $37.21$ .

$f'' (6.1) = 907.924 - 446.52 - 72 \cdot 6.1$

Step 9.2.1.5

Multiply $- 72$ by $6.1$ .

$f'' (6.1) = 907.924 - 446.52 - 439.2$

Step 9.2.2

Simplify by subtracting numbers.

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Step 9.2.2.1

Subtract $446.52$ from $907.924$ .

$f'' (6.1) = 461.404 - 439.2$

Step 9.2.2.2

Subtract $439.2$ from $461.404$ .

$f'' (6.1) = 22.204$

Step 9.2.3

The final answer is $22.204$ .

$22.204$

Step 9.3

At $6.1$ , the second derivative is $22.204$ . Since this is positive, the second derivative is increasing on the interval $(6, \infty)$ .

Increasing on $(6, \infty)$ since $f'' (x) > 0$

Step 10

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 points in this case are $(- 3, \frac{972}{5}), (0, 0), (6, - \frac{11664}{5})$ .

$(- 3, \frac{972}{5}), (0, 0), (6, - \frac{11664}{5})$

Step 11