Calculus Examples

$\frac{1}{10} x^{5} - 12 x^{3}$

Step 1

Write $\frac{1}{10} x^{5} - 12 x^{3}$ as a function.

$f (x) = \frac{1}{10} x^{5} - 12 x^{3}$

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 $\frac{1}{10} x^{5} - 12 x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{10} x^{5}] + \frac{d}{d x} [- 12 x^{3}]$ .

$\frac{d}{d x} [\frac{1}{10} x^{5}] + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2

Evaluate $\frac{d}{d x} [\frac{1}{10} x^{5}]$ .

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Step 2.1.2.1

Since $\frac{1}{10}$ is constant with respect to $x$ , the derivative of $\frac{1}{10} x^{5}$ with respect to $x$ is $\frac{1}{10} \frac{d}{d x} [x^{5}]$ .

$\frac{1}{10} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 12 x^{3}]$

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 $n = 5$ .

$\frac{1}{10} (5 x^{4}) + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2.3

Combine $5$ and $\frac{1}{10}$ .

$\frac{5}{10} x^{4} + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2.4

Combine $\frac{5}{10}$ and $x^{4}$ .

$\frac{5 x^{4}}{10} + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2.5

Cancel the common factor of $5$ and $10$ .

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Step 2.1.2.5.1

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

$\frac{5 (x^{4})}{10} + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2.5.2

Cancel the common factors.

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Step 2.1.2.5.2.1

Factor $5$ out of $10$ .

$\frac{5 x^{4}}{5 \cdot 2} + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2.5.2.2

Cancel the common factor.

$\frac{5 x^{4}}{5 \cdot 2} + \frac{d}{d x} [- 12 x^{3}]$

Step 2.1.2.5.2.3

Rewrite the expression.

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

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 $n = 3$ .

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

Step 2.1.3.3

Multiply $3$ by $- 12$ .

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

Step 2.2

Find the second derivative.

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Step 2.2.1

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

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

Step 2.2.2

Evaluate $\frac{d}{d x} [\frac{x^{4}}{2}]$ .

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Step 2.2.2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{4}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 36 x^{2}]$

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 $n = 4$ .

$\frac{1}{2} (4 x^{3}) + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.2.3

Combine $4$ and $\frac{1}{2}$ .

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

Step 2.2.2.4

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

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

Step 2.2.2.5

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

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Step 2.2.2.5.1

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

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

Step 2.2.2.5.2

Cancel the common factors.

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Step 2.2.2.5.2.1

Factor $2$ out of $2$ .

$\frac{2 (2 x^{3})}{2 (1)} + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.2.5.2.2

Cancel the common factor.

$\frac{2 (2 x^{3})}{2 \cdot 1} + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.2.5.2.3

Rewrite the expression.

$\frac{2 x^{3}}{1} + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.2.5.2.4

Divide $2 x^{3}$ by $1$ .

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

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

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

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 $n = 2$ .

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

Step 2.2.3.3

Multiply $2$ by $- 36$ .

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

Step 2.3

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

$2 x^{3} - 72 x$

Step 3

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

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Step 3.1

Set the second derivative equal to $0$ .

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

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

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

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Step 3.2.1.1

Factor $2 x$ out of $2 x^{3}$ .

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

Step 3.2.1.2

Factor $2 x$ out of $- 72 x$ .

$2 x (x^{2}) + 2 x (- 36) = 0$

Step 3.2.1.3

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 36)$ .

$2 x (x^{2} - 36) = 0$

Step 3.2.2

Rewrite $36$ as $6^{2}$ .

$2 x (x^{2} - 6^{2}) = 0$

Step 3.2.3

Factor.

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Step 3.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$2 x ((x + 6) (x - 6)) = 0$

Step 3.2.3.2

Remove unnecessary parentheses.

$2 x (x + 6) (x - 6) = 0$

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 $0$ .

$x = 0$

$x + 6 = 0$

$x - 6 = 0$

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

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 3.6

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

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Step 3.6.1

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

$x - 6 = 0$

Step 3.6.2

Add $6$ to both sides of the equation.

$x = 6$

Step 3.7

The final solution is all the values that make $2 x (x + 6) (x - 6) = 0$ true.

$x = 0, - 6, 6$

Step 4

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

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Step 4.1

Substitute $0$ in $f (x) = \frac{1}{10} x^{5} - 12 x^{3}$ to find the value of $y$ .

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Step 4.1.1

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

$f (0) = \frac{1}{10} \cdot {(0)}^{5} - 12 {(0)}^{3}$

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 $0$ .

$f (0) = \frac{1}{10} \cdot 0 - 12 {(0)}^{3}$

Step 4.1.2.1.2

Multiply $\frac{1}{10}$ by $0$ .

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

Step 4.1.2.1.3

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

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

Step 4.1.2.1.4

Multiply $- 12$ by $0$ .

$f (0) = 0 + 0$

Step 4.1.2.2

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 $f (x) = \frac{1}{10} x^{5} - 12 x^{3}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4.3

Substitute $- 6$ in $f (x) = \frac{1}{10} x^{5} - 12 x^{3}$ to find the value of $y$ .

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Step 4.3.1

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

$f (- 6) = \frac{1}{10} \cdot {(- 6)}^{5} - 12 {(- 6)}^{3}$

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 $5$ .

$f (- 6) = \frac{1}{10} \cdot - 7776 - 12 {(- 6)}^{3}$

Step 4.3.2.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.2.1

Factor $2$ out of $10$ .

$f (- 6) = \frac{1}{2 (5)} \cdot - 7776 - 12 {(- 6)}^{3}$

Step 4.3.2.1.2.2

Factor $2$ out of $- 7776$ .

$f (- 6) = \frac{1}{2 \cdot 5} \cdot (2 \cdot - 3888) - 12 {(- 6)}^{3}$

Step 4.3.2.1.2.3

Cancel the common factor.

$f (- 6) = \frac{1}{2 \cdot 5} \cdot (2 \cdot - 3888) - 12 {(- 6)}^{3}$

Step 4.3.2.1.2.4

Rewrite the expression.

$f (- 6) = \frac{1}{5} \cdot - 3888 - 12 {(- 6)}^{3}$

Step 4.3.2.1.3

Combine $\frac{1}{5}$ and $- 3888$ .

$f (- 6) = \frac{- 3888}{5} - 12 {(- 6)}^{3}$

Step 4.3.2.1.4

Move the negative in front of the fraction.

$f (- 6) = - \frac{3888}{5} - 12 {(- 6)}^{3}$

Step 4.3.2.1.5

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

$f (- 6) = - \frac{3888}{5} - 12 \cdot - 216$

Step 4.3.2.1.6

Multiply $- 12$ by $- 216$ .

$f (- 6) = - \frac{3888}{5} + 2592$

Step 4.3.2.2

To write $2592$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (- 6) = - \frac{3888}{5} + 2592 \cdot \frac{5}{5}$

Step 4.3.2.3

Combine $2592$ and $\frac{5}{5}$ .

$f (- 6) = - \frac{3888}{5} + \frac{2592 \cdot 5}{5}$

Step 4.3.2.4

Combine the numerators over the common denominator.

$f (- 6) = \frac{- 3888 + 2592 \cdot 5}{5}$

Step 4.3.2.5

Simplify the numerator.

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Step 4.3.2.5.1

Multiply $2592$ by $5$ .

$f (- 6) = \frac{- 3888 + 12960}{5}$

Step 4.3.2.5.2

Add $- 3888$ and $12960$ .

$f (- 6) = \frac{9072}{5}$

Step 4.3.2.6

The final answer is $\frac{9072}{5}$ .

$\frac{9072}{5}$

Step 4.4

The point found by substituting $- 6$ in $f (x) = \frac{1}{10} x^{5} - 12 x^{3}$ is $(- 6, \frac{9072}{5})$ . This point can be an inflection point.

$(- 6, \frac{9072}{5})$

Step 4.5

Substitute $6$ in $f (x) = \frac{1}{10} x^{5} - 12 x^{3}$ to find the value of $y$ .

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Step 4.5.1

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

$f (6) = \frac{1}{10} \cdot {(6)}^{5} - 12 {(6)}^{3}$

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 $6$ to the power of $5$ .

$f (6) = \frac{1}{10} \cdot 7776 - 12 {(6)}^{3}$

Step 4.5.2.1.2

Cancel the common factor of $2$ .

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Step 4.5.2.1.2.1

Factor $2$ out of $10$ .

$f (6) = \frac{1}{2 (5)} \cdot 7776 - 12 {(6)}^{3}$

Step 4.5.2.1.2.2

Factor $2$ out of $7776$ .

$f (6) = \frac{1}{2 \cdot 5} \cdot (2 \cdot 3888) - 12 {(6)}^{3}$

Step 4.5.2.1.2.3

Cancel the common factor.

$f (6) = \frac{1}{2 \cdot 5} \cdot (2 \cdot 3888) - 12 {(6)}^{3}$

Step 4.5.2.1.2.4

Rewrite the expression.

$f (6) = \frac{1}{5} \cdot 3888 - 12 {(6)}^{3}$

Step 4.5.2.1.3

Combine $\frac{1}{5}$ and $3888$ .

$f (6) = \frac{3888}{5} - 12 {(6)}^{3}$

Step 4.5.2.1.4

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

$f (6) = \frac{3888}{5} - 12 \cdot 216$

Step 4.5.2.1.5

Multiply $- 12$ by $216$ .

$f (6) = \frac{3888}{5} - 2592$

Step 4.5.2.2

To write $- 2592$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (6) = \frac{3888}{5} - 2592 \cdot \frac{5}{5}$

Step 4.5.2.3

Combine $- 2592$ and $\frac{5}{5}$ .

$f (6) = \frac{3888}{5} + \frac{- 2592 \cdot 5}{5}$

Step 4.5.2.4

Combine the numerators over the common denominator.

$f (6) = \frac{3888 - 2592 \cdot 5}{5}$

Step 4.5.2.5

Simplify the numerator.

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Step 4.5.2.5.1

Multiply $- 2592$ by $5$ .

$f (6) = \frac{3888 - 12960}{5}$

Step 4.5.2.5.2

Subtract $12960$ from $3888$ .

$f (6) = \frac{- 9072}{5}$

Step 4.5.2.6

Move the negative in front of the fraction.

$f (6) = - \frac{9072}{5}$

Step 4.5.2.7

The final answer is $- \frac{9072}{5}$ .

$- \frac{9072}{5}$

Step 4.6

The point found by substituting $6$ in $f (x) = \frac{1}{10} x^{5} - 12 x^{3}$ is $(6, - \frac{9072}{5})$ . This point can be an inflection point.

$(6, - \frac{9072}{5})$

Step 4.7

Determine the points that could be inflection points.

$(0, 0), (- 6, \frac{9072}{5}), (6, - \frac{9072}{5})$

Step 5

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

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

Step 6

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

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Step 6.1

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

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

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

$f'' (- 6.1) = 2 \cdot - 226.981 - 72 \cdot - 6.1$

Step 6.2.1.2

Multiply $2$ by $- 226.981$ .

$f'' (- 6.1) = - 453.962 - 72 \cdot - 6.1$

Step 6.2.1.3

Multiply $- 72$ by $- 6.1$ .

$f'' (- 6.1) = - 453.962 + 439.2$

Step 6.2.2

Add $- 453.962$ and $439.2$ .

$f'' (- 6.1) = - 14.762$

Step 6.2.3

The final answer is $- 14.762$ .

$- 14.762$

Step 6.3

At $- 6.1$ , the second derivative is $- 14.762$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 6)$

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

Step 7

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

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Step 7.1

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

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

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

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

$f'' (- 3) = 2 \cdot - 27 - 72 \cdot - 3$

Step 7.2.1.2

Multiply $2$ by $- 27$ .

$f'' (- 3) = - 54 - 72 \cdot - 3$

Step 7.2.1.3

Multiply $- 72$ by $- 3$ .

$f'' (- 3) = - 54 + 216$

Step 7.2.2

Add $- 54$ and $216$ .

$f'' (- 3) = 162$

Step 7.2.3

The final answer is $162$ .

$162$

Step 7.3

At $- 3$ , the second derivative is $162$ . Since this is positive, the second derivative is increasing on the interval $(- 6, 0)$ .

Increasing on $(- 6, 0)$ 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.

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

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 $3$ .

$f'' (3) = 2 \cdot 27 - 72 \cdot 3$

Step 8.2.1.2

Multiply $2$ by $27$ .

$f'' (3) = 54 - 72 \cdot 3$

Step 8.2.1.3

Multiply $- 72$ by $3$ .

$f'' (3) = 54 - 216$

Step 8.2.2

Subtract $216$ from $54$ .

$f'' (3) = - 162$

Step 8.2.3

The final answer is $- 162$ .

$- 162$

Step 8.3

At $3$ , the second derivative is $- 162$ . 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.

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

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 $3$ .

$f'' (6.1) = 2 \cdot 226.981 - 72 \cdot 6.1$

Step 9.2.1.2

Multiply $2$ by $226.981$ .

$f'' (6.1) = 453.962 - 72 \cdot 6.1$

Step 9.2.1.3

Multiply $- 72$ by $6.1$ .

$f'' (6.1) = 453.962 - 439.2$

Step 9.2.2

Subtract $439.2$ from $453.962$ .

$f'' (6.1) = 14.762$

Step 9.2.3

The final answer is $14.762$ .

$14.762$

Step 9.3

At $6.1$ , the second derivative is $14.762$ . 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 $(- 6, \frac{9072}{5}), (0, 0), (6, - \frac{9072}{5})$ .

$(- 6, \frac{9072}{5}), (0, 0), (6, - \frac{9072}{5})$

Step 11