Calculus Examples

$\frac{1}{10} x^{5} + 2 x^{4} + 12 x^{3}$

Step 1

Write $\frac{1}{10} x^{5} + 2 x^{4} + 12 x^{3}$ as a function.

$f (x) = \frac{1}{10} x^{5} + 2 x^{4} + 12 x^{3}$

Step 2

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

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

By the Sum Rule, the derivative of $\frac{1}{10} x^{5} + 2 x^{4} + 12 x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{10} x^{5}] + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [12 x^{3}]$ .

$\frac{d}{d x} [\frac{1}{10} x^{5}] + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [12 x^{3}]$

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

$\frac{1}{10} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [12 x^{3}]$

Step 2.1.1.2.2

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

$\frac{1}{10} (5 x^{4}) + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [12 x^{3}]$

Step 2.1.1.2.3

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

$\frac{5}{10} x^{4} + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [12 x^{3}]$

Step 2.1.1.2.4

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

$\frac{5 x^{4}}{10} + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [12 x^{3}]$

Step 2.1.1.2.5

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

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Step 2.1.1.2.5.1

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

$\frac{5 (x^{4})}{10} + \frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [12 x^{3}]$

Step 2.1.1.2.5.2

Cancel the common factors.

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Step 2.1.1.2.5.2.1

Factor $5$ out of $10$ .

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

Step 2.1.1.2.5.2.2

Cancel the common factor.

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

Step 2.1.1.2.5.2.3

Rewrite the expression.

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

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

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 $n = 4$ .

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

Step 2.1.1.3.3

Multiply $4$ by $2$ .

$\frac{x^{4}}{2} + 8 x^{3} + \frac{d}{d x} [12 x^{3}]$

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

$\frac{x^{4}}{2} + 8 x^{3} + 12 \frac{d}{d x} [x^{3}]$

Step 2.1.1.4.2

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

$\frac{x^{4}}{2} + 8 x^{3} + 12 (3 x^{2})$

Step 2.1.1.4.3

Multiply $3$ by $12$ .

$f' (x) = \frac{x^{4}}{2} + 8 x^{3} + 36 x^{2}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $\frac{x^{4}}{2} + 8 x^{3} + 36 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{4}}{2}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [36 x^{2}]$ .

$\frac{d}{d x} [\frac{x^{4}}{2}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$\frac{1}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [36 x^{2}]$

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 $n = 4$ .

$\frac{1}{2} (4 x^{3}) + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.2.3

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

$\frac{4}{2} x^{3} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.2.4

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

$\frac{4 x^{3}}{2} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.2.5

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

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Step 2.1.2.2.5.1

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

$\frac{2 (2 x^{3})}{2} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.2.5.2

Cancel the common factors.

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Step 2.1.2.2.5.2.1

Factor $2$ out of $2$ .

$\frac{2 (2 x^{3})}{2 (1)} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.2.5.2.2

Cancel the common factor.

$\frac{2 (2 x^{3})}{2 \cdot 1} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.2.5.2.3

Rewrite the expression.

$\frac{2 x^{3}}{1} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.2.5.2.4

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

$2 x^{3} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.3

Evaluate $\frac{d}{d x} [8 x^{3}]$ .

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Step 2.1.2.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{3}$ with respect to $x$ is $8 \frac{d}{d x} [x^{3}]$ .

$2 x^{3} + 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.3.2

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

$2 x^{3} + 8 (3 x^{2}) + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.3.3

Multiply $3$ by $8$ .

$2 x^{3} + 24 x^{2} + \frac{d}{d x} [36 x^{2}]$

Step 2.1.2.4

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

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Step 2.1.2.4.1

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

$2 x^{3} + 24 x^{2} + 36 \frac{d}{d x} [x^{2}]$

Step 2.1.2.4.2

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

$2 x^{3} + 24 x^{2} + 36 (2 x)$

Step 2.1.2.4.3

Multiply $2$ by $36$ .

$f'' (x) = 2 x^{3} + 24 x^{2} + 72 x$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $2 x^{3} + 24 x^{2} + 72 x$ .

$2 x^{3} + 24 x^{2} + 72 x$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $2 x^{3} + 24 x^{2} + 72 x = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$2 x^{3} + 24 x^{2} + 72 x = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $2 x$ out of $2 x^{3} + 24 x^{2} + 72 x$ .

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Step 2.2.2.1.1

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

$2 x (x^{2}) + 24 x^{2} + 72 x = 0$

Step 2.2.2.1.2

Factor $2 x$ out of $24 x^{2}$ .

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

Step 2.2.2.1.3

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

$2 x (x^{2}) + 2 x (12 x) + 2 x (36) = 0$

Step 2.2.2.1.4

Factor $2 x$ out of $2 x (x^{2}) + 2 x (12 x)$ .

$2 x (x^{2} + 12 x) + 2 x (36) = 0$

Step 2.2.2.1.5

Factor $2 x$ out of $2 x (x^{2} + 12 x) + 2 x (36)$ .

$2 x (x^{2} + 12 x + 36) = 0$

Step 2.2.2.2

Factor using the perfect square rule.

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Step 2.2.2.2.1

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

$2 x (x^{2} + 12 x + 6^{2}) = 0$

Step 2.2.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 x = 2 \cdot x \cdot 6$

Step 2.2.2.2.3

Rewrite the polynomial.

$2 x (x^{2} + 2 \cdot x \cdot 6 + 6^{2}) = 0$

Step 2.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 6$ .

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

Step 2.2.3

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

$x = 0$

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

Step 2.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.5

Set ${(x + 6)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set ${(x + 6)}^{2}$ equal to $0$ .

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

Step 2.2.5.2

Solve ${(x + 6)}^{2} = 0$ for $x$ .

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Step 2.2.5.2.1

Set the $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 2.2.5.2.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 2.2.6

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

$x = 0, - 6$

Step 3

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:

${x | x \in ℝ}$

Step 4

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

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

Step 5

Substitute any number from the interval $(- \infty, - 6)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 8$ in the expression.

$f'' (- 8) = 2 {(- 8)}^{3} + 24 {(- 8)}^{2} + 72 (- 8)$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 8$ to the power of $3$ .

$f'' (- 8) = 2 \cdot - 512 + 24 {(- 8)}^{2} + 72 (- 8)$

Step 5.2.1.2

Multiply $2$ by $- 512$ .

$f'' (- 8) = - 1024 + 24 {(- 8)}^{2} + 72 (- 8)$

Step 5.2.1.3

Raise $- 8$ to the power of $2$ .

$f'' (- 8) = - 1024 + 24 \cdot 64 + 72 (- 8)$

Step 5.2.1.4

Multiply $24$ by $64$ .

$f'' (- 8) = - 1024 + 1536 + 72 (- 8)$

Step 5.2.1.5

Multiply $72$ by $- 8$ .

$f'' (- 8) = - 1024 + 1536 - 576$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 1024$ and $1536$ .

$f'' (- 8) = 512 - 576$

Step 5.2.2.2

Subtract $576$ from $512$ .

$f'' (- 8) = - 64$

Step 5.2.3

The final answer is $- 64$ .

$- 64$

Step 5.3

The graph is concave down on the interval $(- \infty, - 6)$ because $f'' (- 8)$ is negative.

Concave down on $(- \infty, - 6)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(- 6, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (- 3) = 2 {(- 3)}^{3} + 24 {(- 3)}^{2} + 72 (- 3)$

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

$f'' (- 3) = 2 \cdot - 27 + 24 {(- 3)}^{2} + 72 (- 3)$

Step 6.2.1.2

Multiply $2$ by $- 27$ .

$f'' (- 3) = - 54 + 24 {(- 3)}^{2} + 72 (- 3)$

Step 6.2.1.3

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

$f'' (- 3) = - 54 + 24 \cdot 9 + 72 (- 3)$

Step 6.2.1.4

Multiply $24$ by $9$ .

$f'' (- 3) = - 54 + 216 + 72 (- 3)$

Step 6.2.1.5

Multiply $72$ by $- 3$ .

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

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $- 54$ and $216$ .

$f'' (- 3) = 162 - 216$

Step 6.2.2.2

Subtract $216$ from $162$ .

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

Step 6.2.3

The final answer is $- 54$ .

$- 54$

Step 6.3

The graph is concave down on the interval $(- 6, 0)$ because $f'' (- 3)$ is negative.

Concave down on $(- 6, 0)$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(0, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $2$ in the expression.

$f'' (2) = 2 {(2)}^{3} + 24 {(2)}^{2} + 72 (2)$

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

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

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Step 7.2.1.1.1

Multiply $2$ by ${(2)}^{3}$ .

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Step 7.2.1.1.1.1

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

$f'' (2) = 2 {(2)}^{3} + 24 {(2)}^{2} + 72 (2)$

Step 7.2.1.1.1.2

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

$f'' (2) = 2^{1 + 3} + 24 {(2)}^{2} + 72 (2)$

Step 7.2.1.1.2

Add $1$ and $3$ .

$f'' (2) = 2^{4} + 24 {(2)}^{2} + 72 (2)$

Step 7.2.1.2

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

$f'' (2) = 16 + 24 {(2)}^{2} + 72 (2)$

Step 7.2.1.3

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

$f'' (2) = 16 + 24 \cdot 4 + 72 (2)$

Step 7.2.1.4

Multiply $24$ by $4$ .

$f'' (2) = 16 + 96 + 72 (2)$

Step 7.2.1.5

Multiply $72$ by $2$ .

$f'' (2) = 16 + 96 + 144$

Step 7.2.2

Simplify by adding numbers.

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Step 7.2.2.1

Add $16$ and $96$ .

$f'' (2) = 112 + 144$

Step 7.2.2.2

Add $112$ and $144$ .

$f'' (2) = 256$

Step 7.2.3

The final answer is $256$ .

$256$

Step 7.3

The graph is concave up on the interval $(0, \infty)$ because $f'' (2)$ is positive.

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

Step 8

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

Concave down on $(- \infty, - 6)$ since $f'' (x)$ is negative

Concave down on $(- 6, 0)$ since $f'' (x)$ is negative

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

Step 9