Calculus Examples

$f (x) = \frac{1}{3} x^{6} + 3 x^{5} + \frac{15}{2} x^{4}$

Step 1

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

Tap for more steps...

Step 1.1

Find the second derivative.

Tap for more steps...

Step 1.1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of $\frac{1}{3} x^{6} + 3 x^{5} + \frac{15}{2} x^{4}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{3} x^{6}] + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$ .

$\frac{d}{d x} [\frac{1}{3} x^{6}] + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{6}]$ .

Tap for more steps...

Step 1.1.1.2.1

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

$\frac{1}{3} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$

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 $n = 6$ .

$\frac{1}{3} (6 x^{5}) + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$

Step 1.1.1.2.3

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

$\frac{6}{3} x^{5} + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$

Step 1.1.1.2.4

Combine $\frac{6}{3}$ and $x^{5}$ .

$\frac{6 x^{5}}{3} + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$

Step 1.1.1.2.5

Cancel the common factor of $6$ and $3$ .

Tap for more steps...

Step 1.1.1.2.5.1

Factor $3$ out of $6 x^{5}$ .

$\frac{3 (2 x^{5})}{3} + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$

Step 1.1.1.2.5.2

Cancel the common factors.

Tap for more steps...

Step 1.1.1.2.5.2.1

Factor $3$ out of $3$ .

$\frac{3 (2 x^{5})}{3 (1)} + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$

Step 1.1.1.2.5.2.2

Cancel the common factor.

$\frac{3 (2 x^{5})}{3 \cdot 1} + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$

Step 1.1.1.2.5.2.3

Rewrite the expression.

$\frac{2 x^{5}}{1} + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$

Step 1.1.1.2.5.2.4

Divide $2 x^{5}$ by $1$ .

$2 x^{5} + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$

Step 1.1.1.3

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

Tap for more steps...

Step 1.1.1.3.1

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

$2 x^{5} + 3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{15}{2} x^{4}]$

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 $n = 5$ .

$2 x^{5} + 3 (5 x^{4}) + \frac{d}{d x} [\frac{15}{2} x^{4}]$

Step 1.1.1.3.3

Multiply $5$ by $3$ .

$2 x^{5} + 15 x^{4} + \frac{d}{d x} [\frac{15}{2} x^{4}]$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [\frac{15}{2} x^{4}]$ .

Tap for more steps...

Step 1.1.1.4.1

Since $\frac{15}{2}$ is constant with respect to $x$ , the derivative of $\frac{15}{2} x^{4}$ with respect to $x$ is $\frac{15}{2} \frac{d}{d x} [x^{4}]$ .

$2 x^{5} + 15 x^{4} + \frac{15}{2} \frac{d}{d x} [x^{4}]$

Step 1.1.1.4.2

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

$2 x^{5} + 15 x^{4} + \frac{15}{2} (4 x^{3})$

Step 1.1.1.4.3

Combine $4$ and $\frac{15}{2}$ .

$2 x^{5} + 15 x^{4} + \frac{4 \cdot 15}{2} x^{3}$

Step 1.1.1.4.4

Multiply $4$ by $15$ .

$2 x^{5} + 15 x^{4} + \frac{60}{2} x^{3}$

Step 1.1.1.4.5

Combine $\frac{60}{2}$ and $x^{3}$ .

$2 x^{5} + 15 x^{4} + \frac{60 x^{3}}{2}$

Step 1.1.1.4.6

Cancel the common factor of $60$ and $2$ .

Tap for more steps...

Step 1.1.1.4.6.1

Factor $2$ out of $60 x^{3}$ .

$2 x^{5} + 15 x^{4} + \frac{2 (30 x^{3})}{2}$

Step 1.1.1.4.6.2

Cancel the common factors.

Tap for more steps...

Step 1.1.1.4.6.2.1

Factor $2$ out of $2$ .

$2 x^{5} + 15 x^{4} + \frac{2 (30 x^{3})}{2 (1)}$

Step 1.1.1.4.6.2.2

Cancel the common factor.

$2 x^{5} + 15 x^{4} + \frac{2 (30 x^{3})}{2 \cdot 1}$

Step 1.1.1.4.6.2.3

Rewrite the expression.

$2 x^{5} + 15 x^{4} + \frac{30 x^{3}}{1}$

Step 1.1.1.4.6.2.4

Divide $30 x^{3}$ by $1$ .

$f' (x) = 2 x^{5} + 15 x^{4} + 30 x^{3}$

Step 1.1.2

Find the second derivative.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $2 x^{5} + 15 x^{4} + 30 x^{3}$ with respect to $x$ is $\frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [30 x^{3}]$ .

$\frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [30 x^{3}]$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [2 x^{5}]$ .

Tap for more steps...

Step 1.1.2.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{5}$ with respect to $x$ is $2 \frac{d}{d x} [x^{5}]$ .

$2 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [30 x^{3}]$

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 $n = 5$ .

$2 (5 x^{4}) + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [30 x^{3}]$

Step 1.1.2.2.3

Multiply $5$ by $2$ .

$10 x^{4} + \frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [30 x^{3}]$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [15 x^{4}]$ .

Tap for more steps...

Step 1.1.2.3.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{4}$ with respect to $x$ is $15 \frac{d}{d x} [x^{4}]$ .

$10 x^{4} + 15 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [30 x^{3}]$

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 $n = 4$ .

$10 x^{4} + 15 (4 x^{3}) + \frac{d}{d x} [30 x^{3}]$

Step 1.1.2.3.3

Multiply $4$ by $15$ .

$10 x^{4} + 60 x^{3} + \frac{d}{d x} [30 x^{3}]$

Step 1.1.2.4

Evaluate $\frac{d}{d x} [30 x^{3}]$ .

Tap for more steps...

Step 1.1.2.4.1

Since $30$ is constant with respect to $x$ , the derivative of $30 x^{3}$ with respect to $x$ is $30 \frac{d}{d x} [x^{3}]$ .

$10 x^{4} + 60 x^{3} + 30 \frac{d}{d x} [x^{3}]$

Step 1.1.2.4.2

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

$10 x^{4} + 60 x^{3} + 30 (3 x^{2})$

Step 1.1.2.4.3

Multiply $3$ by $30$ .

$f'' (x) = 10 x^{4} + 60 x^{3} + 90 x^{2}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $10 x^{4} + 60 x^{3} + 90 x^{2}$ .

$10 x^{4} + 60 x^{3} + 90 x^{2}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $10 x^{4} + 60 x^{3} + 90 x^{2} = 0$ .

Tap for more steps...

Step 1.2.1

Set the second derivative equal to $0$ .

$10 x^{4} + 60 x^{3} + 90 x^{2} = 0$

Step 1.2.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.2.1

Factor $10 x^{2}$ out of $10 x^{4} + 60 x^{3} + 90 x^{2}$ .

Tap for more steps...

Step 1.2.2.1.1

Factor $10 x^{2}$ out of $10 x^{4}$ .

$10 x^{2} (x^{2}) + 60 x^{3} + 90 x^{2} = 0$

Step 1.2.2.1.2

Factor $10 x^{2}$ out of $60 x^{3}$ .

$10 x^{2} (x^{2}) + 10 x^{2} (6 x) + 90 x^{2} = 0$

Step 1.2.2.1.3

Factor $10 x^{2}$ out of $90 x^{2}$ .

$10 x^{2} (x^{2}) + 10 x^{2} (6 x) + 10 x^{2} (9) = 0$

Step 1.2.2.1.4

Factor $10 x^{2}$ out of $10 x^{2} (x^{2}) + 10 x^{2} (6 x)$ .

$10 x^{2} (x^{2} + 6 x) + 10 x^{2} (9) = 0$

Step 1.2.2.1.5

Factor $10 x^{2}$ out of $10 x^{2} (x^{2} + 6 x) + 10 x^{2} (9)$ .

$10 x^{2} (x^{2} + 6 x + 9) = 0$

Step 1.2.2.2

Factor using the perfect square rule.

Tap for more steps...

Step 1.2.2.2.1

Rewrite $9$ as $3^{2}$ .

$10 x^{2} (x^{2} + 6 x + 3^{2}) = 0$

Step 1.2.2.2.2

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

$6 x = 2 \cdot x \cdot 3$

Step 1.2.2.2.3

Rewrite the polynomial.

$10 x^{2} (x^{2} + 2 \cdot x \cdot 3 + 3^{2}) = 0$

Step 1.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

$10 x^{2} {(x + 3)}^{2} = 0$

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 $0$ .

$x^{2} = 0$

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

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$x^{2} = 0$

Step 1.2.4.2

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

Tap for more steps...

Step 1.2.4.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.4.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 1.2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.5

Set ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set ${(x + 3)}^{2}$ equal to $0$ .

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

Step 1.2.5.2

Solve ${(x + 3)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 1.2.5.2.1

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

$x + 3 = 0$

Step 1.2.5.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.6

The final solution is all the values that make $10 x^{2} {(x + 3)}^{2} = 0$ true.

$x = 0, - 3$

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.

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

Step 4

Substitute any number from the interval $(- \infty, - 3)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 4.1

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

$f'' (- 6) = 10 {(- 6)}^{4} + 60 {(- 6)}^{3} + 90 {(- 6)}^{2}$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

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

$f'' (- 6) = 10 \cdot 1296 + 60 {(- 6)}^{3} + 90 {(- 6)}^{2}$

Step 4.2.1.2

Multiply $10$ by $1296$ .

$f'' (- 6) = 12960 + 60 {(- 6)}^{3} + 90 {(- 6)}^{2}$

Step 4.2.1.3

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

$f'' (- 6) = 12960 + 60 \cdot - 216 + 90 {(- 6)}^{2}$

Step 4.2.1.4

Multiply $60$ by $- 216$ .

$f'' (- 6) = 12960 - 12960 + 90 {(- 6)}^{2}$

Step 4.2.1.5

Raise $- 6$ to the power of $2$ .

$f'' (- 6) = 12960 - 12960 + 90 \cdot 36$

Step 4.2.1.6

Multiply $90$ by $36$ .

$f'' (- 6) = 12960 - 12960 + 3240$

Step 4.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.2.1

Subtract $12960$ from $12960$ .

$f'' (- 6) = 0 + 3240$

Step 4.2.2.2

Add $0$ and $3240$ .

$f'' (- 6) = 3240$

Step 4.2.3

The final answer is $3240$ .

$3240$

Step 4.3

The graph is concave up on the interval $(- \infty, - 3)$ because $f'' (- 6)$ is positive.

Concave up on $(- \infty, - 3)$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(- 3, 0)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 5.1

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

$f'' (- 2) = 10 {(- 2)}^{4} + 60 {(- 2)}^{3} + 90 {(- 2)}^{2}$

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

$f'' (- 2) = 10 \cdot 16 + 60 {(- 2)}^{3} + 90 {(- 2)}^{2}$

Step 5.2.1.2

Multiply $10$ by $16$ .

$f'' (- 2) = 160 + 60 {(- 2)}^{3} + 90 {(- 2)}^{2}$

Step 5.2.1.3

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

$f'' (- 2) = 160 + 60 \cdot - 8 + 90 {(- 2)}^{2}$

Step 5.2.1.4

Multiply $60$ by $- 8$ .

$f'' (- 2) = 160 - 480 + 90 {(- 2)}^{2}$

Step 5.2.1.5

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

$f'' (- 2) = 160 - 480 + 90 \cdot 4$

Step 5.2.1.6

Multiply $90$ by $4$ .

$f'' (- 2) = 160 - 480 + 360$

Step 5.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 5.2.2.1

Subtract $480$ from $160$ .

$f'' (- 2) = - 320 + 360$

Step 5.2.2.2

Add $- 320$ and $360$ .

$f'' (- 2) = 40$

Step 5.2.3

The final answer is $40$ .

$40$

Step 5.3

The graph is concave up on the interval $(- 3, 0)$ because $f'' (- 2)$ is positive.

Concave up on $(- 3, 0)$ since $f'' (x)$ is positive

Step 6

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

Tap for more steps...

Step 6.1

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

$f'' (2) = 10 {(2)}^{4} + 60 {(2)}^{3} + 90 {(2)}^{2}$

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 $2$ to the power of $4$ .

$f'' (2) = 10 \cdot 16 + 60 {(2)}^{3} + 90 {(2)}^{2}$

Step 6.2.1.2

Multiply $10$ by $16$ .

$f'' (2) = 160 + 60 {(2)}^{3} + 90 {(2)}^{2}$

Step 6.2.1.3

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

$f'' (2) = 160 + 60 \cdot 8 + 90 {(2)}^{2}$

Step 6.2.1.4

Multiply $60$ by $8$ .

$f'' (2) = 160 + 480 + 90 {(2)}^{2}$

Step 6.2.1.5

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

$f'' (2) = 160 + 480 + 90 \cdot 4$

Step 6.2.1.6

Multiply $90$ by $4$ .

$f'' (2) = 160 + 480 + 360$

Step 6.2.2

Simplify by adding numbers.

Tap for more steps...

Step 6.2.2.1

Add $160$ and $480$ .

$f'' (2) = 640 + 360$

Step 6.2.2.2

Add $640$ and $360$ .

$f'' (2) = 1000$

Step 6.2.3

The final answer is $1000$ .

$1000$

Step 6.3

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 7

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

Concave up on $(- \infty, - 3)$ since $f'' (x)$ is positive

Concave up on $(- 3, 0)$ since $f'' (x)$ is positive

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

Step 8