Calculus Examples

$f (x) = 3 x^{4} - 16 x^{3} + 18 x^{2}$

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 $3 x^{4} - 16 x^{3} + 18 x^{2}$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 16 x^{3}] + \frac{d}{d x} [18 x^{2}]$ .

$f' (x) = \frac{d}{d x} (3 x^{4}) + \frac{d}{d x} (- 16 x^{3}) + \frac{d}{d x} (18 x^{2})$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$f' (x) = 3 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 16 x^{3}) + \frac{d}{d x} (18 x^{2})$

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 $n = 4$ .

$f' (x) = 3 (4 x^{3}) + \frac{d}{d x} (- 16 x^{3}) + \frac{d}{d x} (18 x^{2})$

Step 1.1.1.2.3

Multiply $4$ by $3$ .

$f' (x) = 12 x^{3} + \frac{d}{d x} (- 16 x^{3}) + \frac{d}{d x} (18 x^{2})$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- 16 x^{3}]$ .

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Step 1.1.1.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 x^{3}$ with respect to $x$ is $- 16 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 12 x^{3} - 16 \frac{d}{d x} x^{3} + \frac{d}{d x} (18 x^{2})$

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 $n = 3$ .

$f' (x) = 12 x^{3} - 16 (3 x^{2}) + \frac{d}{d x} (18 x^{2})$

Step 1.1.1.3.3

Multiply $3$ by $- 16$ .

$f' (x) = 12 x^{3} - 48 x^{2} + \frac{d}{d x} (18 x^{2})$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [18 x^{2}]$ .

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Step 1.1.1.4.1

Since $18$ is constant with respect to $x$ , the derivative of $18 x^{2}$ with respect to $x$ is $18 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 12 x^{3} - 48 x^{2} + 18 \frac{d}{d x} (x^{2})$

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 $n = 2$ .

$f' (x) = 12 x^{3} - 48 x^{2} + 18 (2 x)$

Step 1.1.1.4.3

Multiply $2$ by $18$ .

$f' (x) = 12 x^{3} - 48 x^{2} + 36 x$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $12 x^{3} - 48 x^{2} + 36 x$ with respect to $x$ is $\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [36 x]$ .

$f'' (x) = \frac{d}{d x} (12 x^{3}) + \frac{d}{d x} (- 48 x^{2}) + \frac{d}{d x} (36 x)$

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

$f'' (x) = 12 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 48 x^{2}) + \frac{d}{d x} (36 x)$

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 $n = 3$ .

$f'' (x) = 12 (3 x^{2}) + \frac{d}{d x} (- 48 x^{2}) + \frac{d}{d x} (36 x)$

Step 1.1.2.2.3

Multiply $3$ by $12$ .

$f'' (x) = 36 x^{2} + \frac{d}{d x} (- 48 x^{2}) + \frac{d}{d x} (36 x)$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- 48 x^{2}]$ .

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Step 1.1.2.3.1

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48 x^{2}$ with respect to $x$ is $- 48 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 36 x^{2} - 48 \frac{d}{d x} x^{2} + \frac{d}{d x} (36 x)$

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 $n = 2$ .

$f'' (x) = 36 x^{2} - 48 (2 x) + \frac{d}{d x} (36 x)$

Step 1.1.2.3.3

Multiply $2$ by $- 48$ .

$f'' (x) = 36 x^{2} - 96 x + \frac{d}{d x} (36 x)$

Step 1.1.2.4

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

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Step 1.1.2.4.1

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

$f'' (x) = 36 x^{2} - 96 x + 36 \frac{d}{d x} (x)$

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 $n = 1$ .

$f'' (x) = 36 x^{2} - 96 x + 36 \cdot 1$

Step 1.1.2.4.3

Multiply $36$ by $1$ .

$f'' (x) = 36 x^{2} - 96 x + 36$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $36 x^{2} - 96 x + 36$ .

$36 x^{2} - 96 x + 36$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $36 x^{2} - 96 x + 36 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$36 x^{2} - 96 x + 36 = 0$

Step 1.2.2

Factor $12$ out of $36 x^{2} - 96 x + 36$ .

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Step 1.2.2.1

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

$12 (3 x^{2}) - 96 x + 36 = 0$

Step 1.2.2.2

Factor $12$ out of $- 96 x$ .

$12 (3 x^{2}) + 12 (- 8 x) + 36 = 0$

Step 1.2.2.3

Factor $12$ out of $36$ .

$12 (3 x^{2}) + 12 (- 8 x) + 12 (3) = 0$

Step 1.2.2.4

Factor $12$ out of $12 (3 x^{2}) + 12 (- 8 x)$ .

$12 (3 x^{2} - 8 x) + 12 (3) = 0$

Step 1.2.2.5

Factor $12$ out of $12 (3 x^{2} - 8 x) + 12 (3)$ .

$12 (3 x^{2} - 8 x + 3) = 0$

Step 1.2.3

Divide each term in $12 (3 x^{2} - 8 x + 3) = 0$ by $12$ and simplify.

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Step 1.2.3.1

Divide each term in $12 (3 x^{2} - 8 x + 3) = 0$ by $12$ .

$\frac{12 (3 x^{2} - 8 x + 3)}{12} = \frac{0}{12}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $12$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{12 (3 x^{2} - 8 x + 3)}{12} = \frac{0}{12}$

Step 1.2.3.2.1.2

Divide $3 x^{2} - 8 x + 3$ by $1$ .

$3 x^{2} - 8 x + 3 = \frac{0}{12}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $12$ .

$3 x^{2} - 8 x + 3 = 0$

Step 1.2.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.5

Substitute the values $a = 3$ , $b = - 8$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (3 \cdot 3)}}{2 \cdot 3}$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot 3}}{2 \cdot 3}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 3 \cdot 3$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot 3}}{2 \cdot 3}$

Step 1.2.6.1.2.2

Multiply $- 12$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 36}}{2 \cdot 3}$

Step 1.2.6.1.3

Subtract $36$ from $64$ .

$x = \frac{8 \pm \sqrt{28}}{2 \cdot 3}$

Step 1.2.6.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 1.2.6.1.4.1

Factor $4$ out of $28$ .

$x = \frac{8 \pm \sqrt{4 (7)}}{2 \cdot 3}$

Step 1.2.6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{8 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 3}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{7}}{2 \cdot 3}$

Step 1.2.6.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 2 \sqrt{7}}{6}$

Step 1.2.6.3

Simplify $\frac{8 \pm 2 \sqrt{7}}{6}$ .

$x = \frac{4 \pm \sqrt{7}}{3}$

Step 1.2.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot 3}}{2 \cdot 3}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 3 \cdot 3$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot 3}}{2 \cdot 3}$

Step 1.2.7.1.2.2

Multiply $- 12$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 36}}{2 \cdot 3}$

Step 1.2.7.1.3

Subtract $36$ from $64$ .

$x = \frac{8 \pm \sqrt{28}}{2 \cdot 3}$

Step 1.2.7.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 1.2.7.1.4.1

Factor $4$ out of $28$ .

$x = \frac{8 \pm \sqrt{4 (7)}}{2 \cdot 3}$

Step 1.2.7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{8 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 3}$

Step 1.2.7.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{7}}{2 \cdot 3}$

Step 1.2.7.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 2 \sqrt{7}}{6}$

Step 1.2.7.3

Simplify $\frac{8 \pm 2 \sqrt{7}}{6}$ .

$x = \frac{4 \pm \sqrt{7}}{3}$

Step 1.2.7.4

Change the $\pm$ to $+$ .

$x = \frac{4 + \sqrt{7}}{3}$

Step 1.2.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.8.1

Simplify the numerator.

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Step 1.2.8.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot 3}}{2 \cdot 3}$

Step 1.2.8.1.2

Multiply $- 4 \cdot 3 \cdot 3$ .

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Step 1.2.8.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot 3}}{2 \cdot 3}$

Step 1.2.8.1.2.2

Multiply $- 12$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 36}}{2 \cdot 3}$

Step 1.2.8.1.3

Subtract $36$ from $64$ .

$x = \frac{8 \pm \sqrt{28}}{2 \cdot 3}$

Step 1.2.8.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 1.2.8.1.4.1

Factor $4$ out of $28$ .

$x = \frac{8 \pm \sqrt{4 (7)}}{2 \cdot 3}$

Step 1.2.8.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{8 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 3}$

Step 1.2.8.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{7}}{2 \cdot 3}$

Step 1.2.8.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 2 \sqrt{7}}{6}$

Step 1.2.8.3

Simplify $\frac{8 \pm 2 \sqrt{7}}{6}$ .

$x = \frac{4 \pm \sqrt{7}}{3}$

Step 1.2.8.4

Change the $\pm$ to $-$ .

$x = \frac{4 - \sqrt{7}}{3}$

Step 1.2.9

The final answer is the combination of both solutions.

$x = \frac{4 + \sqrt{7}}{3}, \frac{4 - \sqrt{7}}{3}$

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.

$(- \infty, \frac{4 - \sqrt{7}}{3}) \cup (\frac{4 - \sqrt{7}}{3}, \frac{4 + \sqrt{7}}{3}) \cup (\frac{4 + \sqrt{7}}{3}, \infty)$

Step 4

Substitute any number from the interval $(- \infty, \frac{4 - \sqrt{7}}{3})$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

$f'' (0) = 36 {(0)}^{2} - 96 \cdot 0 + 36$

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

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

$f'' (0) = 36 \cdot 0 - 96 \cdot 0 + 36$

Step 4.2.1.2

Multiply $36$ by $0$ .

$f'' (0) = 0 - 96 \cdot 0 + 36$

Step 4.2.1.3

Multiply $- 96$ by $0$ .

$f'' (0) = 0 + 0 + 36$

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add $0$ and $0$ .

$f'' (0) = 0 + 36$

Step 4.2.2.2

Add $0$ and $36$ .

$f'' (0) = 36$

Step 4.2.3

The final answer is $36$ .

$36$

Step 4.3

The graph is concave up on the interval $(- \infty, \frac{4 - \sqrt{7}}{3})$ because $f'' (0)$ is positive.

Concave up on $(- \infty, \frac{4 - \sqrt{7}}{3})$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(\frac{4 - \sqrt{7}}{3}, \frac{4 + \sqrt{7}}{3})$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $1$ in the expression.

$f'' (1) = 36 {(1)}^{2} - 96 \cdot 1 + 36$

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

One to any power is one.

$f'' (1) = 36 \cdot 1 - 96 \cdot 1 + 36$

Step 5.2.1.2

Multiply $36$ by $1$ .

$f'' (1) = 36 - 96 \cdot 1 + 36$

Step 5.2.1.3

Multiply $- 96$ by $1$ .

$f'' (1) = 36 - 96 + 36$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $96$ from $36$ .

$f'' (1) = - 60 + 36$

Step 5.2.2.2

Add $- 60$ and $36$ .

$f'' (1) = - 24$

Step 5.2.3

The final answer is $- 24$ .

$- 24$

Step 5.3

The graph is concave down on the interval $(\frac{4 - \sqrt{7}}{3}, \frac{4 + \sqrt{7}}{3})$ because $f'' (1)$ is negative.

Concave down on $(\frac{4 - \sqrt{7}}{3}, \frac{4 + \sqrt{7}}{3})$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(\frac{4 + \sqrt{7}}{3}, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $5$ in the expression.

$f'' (5) = 36 {(5)}^{2} - 96 \cdot 5 + 36$

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 $5$ to the power of $2$ .

$f'' (5) = 36 \cdot 25 - 96 \cdot 5 + 36$

Step 6.2.1.2

Multiply $36$ by $25$ .

$f'' (5) = 900 - 96 \cdot 5 + 36$

Step 6.2.1.3

Multiply $- 96$ by $5$ .

$f'' (5) = 900 - 480 + 36$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $480$ from $900$ .

$f'' (5) = 420 + 36$

Step 6.2.2.2

Add $420$ and $36$ .

$f'' (5) = 456$

Step 6.2.3

The final answer is $456$ .

$456$

Step 6.3

The graph is concave up on the interval $(\frac{4 + \sqrt{7}}{3}, \infty)$ because $f'' (5)$ is positive.

Concave up on $(\frac{4 + \sqrt{7}}{3}, \infty)$ 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 $(- \infty, \frac{4 - \sqrt{7}}{3})$ since $f'' (x)$ is positive

Concave down on $(\frac{4 - \sqrt{7}}{3}, \frac{4 + \sqrt{7}}{3})$ since $f'' (x)$ is negative

Concave up on $(\frac{4 + \sqrt{7}}{3}, \infty)$ since $f'' (x)$ is positive

Step 8