Calculus Examples

$f (x) = \frac{3}{20} x^{5} - 7 x^{3}$

Step 1

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

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of $\frac{3}{20} x^{5} - 7 x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{20} x^{5}] + \frac{d}{d x} [- 7 x^{3}]$ .

$\frac{d}{d x} [\frac{3}{20} x^{5}] + \frac{d}{d x} [- 7 x^{3}]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [\frac{3}{20} x^{5}]$ .

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Step 1.1.1.2.1

Since $\frac{3}{20}$ is constant with respect to $x$ , the derivative of $\frac{3}{20} x^{5}$ with respect to $x$ is $\frac{3}{20} \frac{d}{d x} [x^{5}]$ .

$\frac{3}{20} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 7 x^{3}]$

Step 1.1.1.2.2

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

$\frac{3}{20} (5 x^{4}) + \frac{d}{d x} [- 7 x^{3}]$

Step 1.1.1.2.3

Combine $5$ and $\frac{3}{20}$ .

$\frac{5 \cdot 3}{20} x^{4} + \frac{d}{d x} [- 7 x^{3}]$

Step 1.1.1.2.4

Multiply $5$ by $3$ .

$\frac{15}{20} x^{4} + \frac{d}{d x} [- 7 x^{3}]$

Step 1.1.1.2.5

Combine $\frac{15}{20}$ and $x^{4}$ .

$\frac{15 x^{4}}{20} + \frac{d}{d x} [- 7 x^{3}]$

Step 1.1.1.2.6

Cancel the common factor of $15$ and $20$ .

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Step 1.1.1.2.6.1

Factor $5$ out of $15 x^{4}$ .

$\frac{5 (3 x^{4})}{20} + \frac{d}{d x} [- 7 x^{3}]$

Step 1.1.1.2.6.2

Cancel the common factors.

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Step 1.1.1.2.6.2.1

Factor $5$ out of $20$ .

$\frac{5 (3 x^{4})}{5 (4)} + \frac{d}{d x} [- 7 x^{3}]$

Step 1.1.1.2.6.2.2

Cancel the common factor.

$\frac{5 (3 x^{4})}{5 \cdot 4} + \frac{d}{d x} [- 7 x^{3}]$

Step 1.1.1.2.6.2.3

Rewrite the expression.

$\frac{3 x^{4}}{4} + \frac{d}{d x} [- 7 x^{3}]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- 7 x^{3}]$ .

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Step 1.1.1.3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{3}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{3}]$ .

$\frac{3 x^{4}}{4} - 7 \frac{d}{d x} [x^{3}]$

Step 1.1.1.3.2

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

$\frac{3 x^{4}}{4} - 7 (3 x^{2})$

Step 1.1.1.3.3

Multiply $3$ by $- 7$ .

$f' (x) = \frac{3 x^{4}}{4} - 21 x^{2}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $\frac{3 x^{4}}{4} - 21 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{3 x^{4}}{4}] + \frac{d}{d x} [- 21 x^{2}]$ .

$\frac{d}{d x} [\frac{3 x^{4}}{4}] + \frac{d}{d x} [- 21 x^{2}]$

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

$\frac{3}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 21 x^{2}]$

Step 1.1.2.2.2

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

$\frac{3}{4} (4 x^{3}) + \frac{d}{d x} [- 21 x^{2}]$

Step 1.1.2.2.3

Combine $4$ and $\frac{3}{4}$ .

$\frac{4 \cdot 3}{4} x^{3} + \frac{d}{d x} [- 21 x^{2}]$

Step 1.1.2.2.4

Multiply $4$ by $3$ .

$\frac{12}{4} x^{3} + \frac{d}{d x} [- 21 x^{2}]$

Step 1.1.2.2.5

Combine $\frac{12}{4}$ and $x^{3}$ .

$\frac{12 x^{3}}{4} + \frac{d}{d x} [- 21 x^{2}]$

Step 1.1.2.2.6

Cancel the common factor of $12$ and $4$ .

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Step 1.1.2.2.6.1

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

$\frac{4 (3 x^{3})}{4} + \frac{d}{d x} [- 21 x^{2}]$

Step 1.1.2.2.6.2

Cancel the common factors.

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Step 1.1.2.2.6.2.1

Factor $4$ out of $4$ .

$\frac{4 (3 x^{3})}{4 (1)} + \frac{d}{d x} [- 21 x^{2}]$

Step 1.1.2.2.6.2.2

Cancel the common factor.

$\frac{4 (3 x^{3})}{4 \cdot 1} + \frac{d}{d x} [- 21 x^{2}]$

Step 1.1.2.2.6.2.3

Rewrite the expression.

$\frac{3 x^{3}}{1} + \frac{d}{d x} [- 21 x^{2}]$

Step 1.1.2.2.6.2.4

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

$3 x^{3} + \frac{d}{d x} [- 21 x^{2}]$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- 21 x^{2}]$ .

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Step 1.1.2.3.1

Since $- 21$ is constant with respect to $x$ , the derivative of $- 21 x^{2}$ with respect to $x$ is $- 21 \frac{d}{d x} [x^{2}]$ .

$3 x^{3} - 21 \frac{d}{d x} [x^{2}]$

Step 1.1.2.3.2

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

$3 x^{3} - 21 (2 x)$

Step 1.1.2.3.3

Multiply $2$ by $- 21$ .

$f'' (x) = 3 x^{3} - 42 x$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $3 x^{3} - 42 x$ .

$3 x^{3} - 42 x$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $3 x^{3} - 42 x = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$3 x^{3} - 42 x = 0$

Step 1.2.2

Factor $3 x$ out of $3 x^{3} - 42 x$ .

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Step 1.2.2.1

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

$3 x (x^{2}) - 42 x = 0$

Step 1.2.2.2

Factor $3 x$ out of $- 42 x$ .

$3 x (x^{2}) + 3 x (- 14) = 0$

Step 1.2.2.3

Factor $3 x$ out of $3 x (x^{2}) + 3 x (- 14)$ .

$3 x (x^{2} - 14) = 0$

Step 1.2.3

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

$x = 0$

$x^{2} - 14 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $x^{2} - 14$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x^{2} - 14$ equal to $0$ .

$x^{2} - 14 = 0$

Step 1.2.5.2

Solve $x^{2} - 14 = 0$ for $x$ .

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Step 1.2.5.2.1

Add $14$ to both sides of the equation.

$x^{2} = 14$

Step 1.2.5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{14}$

Step 1.2.5.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.5.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{14}$

Step 1.2.5.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{14}$

Step 1.2.5.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{14}, - \sqrt{14}$

Step 1.2.6

The final solution is all the values that make $3 x (x^{2} - 14) = 0$ true.

$x = 0, \sqrt{14}, - \sqrt{14}$

Step 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:

${x | x \in ℝ}$

Step 3

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

$(- \infty, - \sqrt{14}) \cup (- \sqrt{14}, 0) \cup (0, \sqrt{14}) \cup (\sqrt{14}, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - \sqrt{14})$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

$f'' (- 6) = 3 {(- 6)}^{3} - 42 \cdot - 6$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$f'' (- 6) = 3 \cdot - 216 - 42 \cdot - 6$

Step 4.2.1.2

Multiply $3$ by $- 216$ .

$f'' (- 6) = - 648 - 42 \cdot - 6$

Step 4.2.1.3

Multiply $- 42$ by $- 6$ .

$f'' (- 6) = - 648 + 252$

Step 4.2.2

Add $- 648$ and $252$ .

$f'' (- 6) = - 396$

Step 4.2.3

The final answer is $- 396$ .

$- 396$

Step 4.3

The graph is concave down on the interval $(- \infty, - \sqrt{14})$ because $f'' (- 6)$ is negative.

Concave down on $(- \infty, - \sqrt{14})$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(- \sqrt{14}, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (- 2) = 3 {(- 2)}^{3} - 42 \cdot - 2$

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

$f'' (- 2) = 3 \cdot - 8 - 42 \cdot - 2$

Step 5.2.1.2

Multiply $3$ by $- 8$ .

$f'' (- 2) = - 24 - 42 \cdot - 2$

Step 5.2.1.3

Multiply $- 42$ by $- 2$ .

$f'' (- 2) = - 24 + 84$

Step 5.2.2

Add $- 24$ and $84$ .

$f'' (- 2) = 60$

Step 5.2.3

The final answer is $60$ .

$60$

Step 5.3

The graph is concave up on the interval $(- \sqrt{14}, 0)$ because $f'' (- 2)$ is positive.

Concave up on $(- \sqrt{14}, 0)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(0, \sqrt{14})$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (2) = 3 {(2)}^{3} - 42 \cdot 2$

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 $2$ to the power of $3$ .

$f'' (2) = 3 \cdot 8 - 42 \cdot 2$

Step 6.2.1.2

Multiply $3$ by $8$ .

$f'' (2) = 24 - 42 \cdot 2$

Step 6.2.1.3

Multiply $- 42$ by $2$ .

$f'' (2) = 24 - 84$

Step 6.2.2

Subtract $84$ from $24$ .

$f'' (2) = - 60$

Step 6.2.3

The final answer is $- 60$ .

$- 60$

Step 6.3

The graph is concave down on the interval $(0, \sqrt{14})$ because $f'' (2)$ is negative.

Concave down on $(0, \sqrt{14})$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(\sqrt{14}, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

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

$f'' (6) = 3 {(6)}^{3} - 42 \cdot 6$

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 $6$ to the power of $3$ .

$f'' (6) = 3 \cdot 216 - 42 \cdot 6$

Step 7.2.1.2

Multiply $3$ by $216$ .

$f'' (6) = 648 - 42 \cdot 6$

Step 7.2.1.3

Multiply $- 42$ by $6$ .

$f'' (6) = 648 - 252$

Step 7.2.2

Subtract $252$ from $648$ .

$f'' (6) = 396$

Step 7.2.3

The final answer is $396$ .

$396$

Step 7.3

The graph is concave up on the interval $(\sqrt{14}, \infty)$ because $f'' (6)$ is positive.

Concave up on $(\sqrt{14}, \infty)$ 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 $(- \infty, - \sqrt{14})$ since $f'' (x)$ is negative

Concave up on $(- \sqrt{14}, 0)$ since $f'' (x)$ is positive

Concave down on $(0, \sqrt{14})$ since $f'' (x)$ is negative

Concave up on $(\sqrt{14}, \infty)$ since $f'' (x)$ is positive

Step 9