Calculus Examples

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

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 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 1.1.2

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

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Step 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 1.1.2.2

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 1.1.2.3

Multiply $4$ by $3$ .

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

Step 1.1.3

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

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Step 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 1.1.3.2

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 1.1.3.3

Multiply $3$ by $- 16$ .

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

Step 1.1.4

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

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Step 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 1.1.4.2

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 1.1.4.3

Multiply $2$ by $18$ .

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

Step 1.2

Find the second derivative.

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Step 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 1.2.2

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

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Step 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 1.2.2.2

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 1.2.2.3

Multiply $3$ by $12$ .

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

Step 1.2.3

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

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Step 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 1.2.3.2

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 1.2.3.3

Multiply $2$ by $- 48$ .

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

Step 1.2.4

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

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Step 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 1.2.4.2

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 1.2.4.3

Multiply $36$ by $1$ .

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

Step 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 2

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

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Step 2.1

Set the second derivative equal to $0$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Factor $12$ out of $36$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.3

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

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Step 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 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $12$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

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

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $12$ .

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

Step 2.4

Use the quadratic formula to find the solutions.

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

Step 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 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 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 2.6.1.2

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

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Step 2.6.1.2.1

Multiply $- 4$ by $3$ .

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

Step 2.6.1.2.2

Multiply $- 12$ by $3$ .

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

Step 2.6.1.3

Subtract $36$ from $64$ .

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

Step 2.6.1.4

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

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Step 2.6.1.4.1

Factor $4$ out of $28$ .

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

Step 2.6.1.4.2

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

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

Step 2.6.1.5

Pull terms out from under the radical.

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

Step 2.6.2

Multiply $2$ by $3$ .

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

Step 2.6.3

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

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

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 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 2.7.1.2

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

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Step 2.7.1.2.1

Multiply $- 4$ by $3$ .

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

Step 2.7.1.2.2

Multiply $- 12$ by $3$ .

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

Step 2.7.1.3

Subtract $36$ from $64$ .

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

Step 2.7.1.4

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

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Step 2.7.1.4.1

Factor $4$ out of $28$ .

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

Step 2.7.1.4.2

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

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

Step 2.7.1.5

Pull terms out from under the radical.

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

Step 2.7.2

Multiply $2$ by $3$ .

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

Step 2.7.3

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

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

Step 2.7.4

Change the $\pm$ to $+$ .

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

Step 2.8

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

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Step 2.8.1

Simplify the numerator.

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Step 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 2.8.1.2

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

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Step 2.8.1.2.1

Multiply $- 4$ by $3$ .

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

Step 2.8.1.2.2

Multiply $- 12$ by $3$ .

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

Step 2.8.1.3

Subtract $36$ from $64$ .

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

Step 2.8.1.4

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

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Step 2.8.1.4.1

Factor $4$ out of $28$ .

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

Step 2.8.1.4.2

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

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

Step 2.8.1.5

Pull terms out from under the radical.

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

Step 2.8.2

Multiply $2$ by $3$ .

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

Step 2.8.3

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

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

Step 2.8.4

Change the $\pm$ to $-$ .

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

Step 2.9

The final answer is the combination of both solutions.

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

Step 3

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

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Step 3.1

Substitute $2.21525043$ in $f (x) = 3 x^{4} - 16 x^{3} + 18 x^{2}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $2.21525043$ in the expression.

$f (2.21525043) = 3 {(2.21525043)}^{4} - 16 {(2.21525043)}^{3} + 18 {(2.21525043)}^{2}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Raise $2.21525043$ to the power of $4$ .

$f (2.21525043) = 3 \cdot 24.08193188 - 16 {(2.21525043)}^{3} + 18 {(2.21525043)}^{2}$

Step 3.1.2.1.2

Multiply $3$ by $24.08193188$ .

$f (2.21525043) = 72.24579564 - 16 {(2.21525043)}^{3} + 18 {(2.21525043)}^{2}$

Step 3.1.2.1.3

Raise $2.21525043$ to the power of $3$ .

$f (2.21525043) = 72.24579564 - 16 \cdot 10.87097489 + 18 {(2.21525043)}^{2}$

Step 3.1.2.1.4

Multiply $- 16$ by $10.87097489$ .

$f (2.21525043) = 72.24579564 - 173.93559828 + 18 {(2.21525043)}^{2}$

Step 3.1.2.1.5

Raise $2.21525043$ to the power of $2$ .

$f (2.21525043) = 72.24579564 - 173.93559828 + 18 \cdot 4.90733449$

Step 3.1.2.1.6

Multiply $18$ by $4.90733449$ .

$f (2.21525043) = 72.24579564 - 173.93559828 + 88.33202097$

Step 3.1.2.2

Simplify by adding and subtracting.

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Step 3.1.2.2.1

Subtract $173.93559828$ from $72.24579564$ .

$f (2.21525043) = - 101.68980263 + 88.33202097$

Step 3.1.2.2.2

Add $- 101.68980263$ and $88.33202097$ .

$f (2.21525043) = - 13.35778166$

Step 3.1.2.3

The final answer is $- 13.35778166$ .

$- 13.35778166$

Step 3.2

The point found by substituting $2.21525043$ in $f (x) = 3 x^{4} - 16 x^{3} + 18 x^{2}$ is $(2.21525043, - 13.35778166)$ . This point can be an inflection point.

$(2.21525043, - 13.35778166)$

Step 3.3

Substitute $0.45141622$ in $f (x) = 3 x^{4} - 16 x^{3} + 18 x^{2}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $0.45141622$ in the expression.

$f (0.45141622) = 3 {(0.45141622)}^{4} - 16 {(0.45141622)}^{3} + 18 {(0.45141622)}^{2}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Raise $0.45141622$ to the power of $4$ .

$f (0.45141622) = 3 \cdot 0.0415249 - 16 {(0.45141622)}^{3} + 18 {(0.45141622)}^{2}$

Step 3.3.2.1.2

Multiply $3$ by $0.0415249$ .

$f (0.45141622) = 0.12457472 - 16 {(0.45141622)}^{3} + 18 {(0.45141622)}^{2}$

Step 3.3.2.1.3

Raise $0.45141622$ to the power of $3$ .

$f (0.45141622) = 0.12457472 - 16 \cdot 0.09198807 + 18 {(0.45141622)}^{2}$

Step 3.3.2.1.4

Multiply $- 16$ by $0.09198807$ .

$f (0.45141622) = 0.12457472 - 1.47180912 + 18 {(0.45141622)}^{2}$

Step 3.3.2.1.5

Raise $0.45141622$ to the power of $2$ .

$f (0.45141622) = 0.12457472 - 1.47180912 + 18 \cdot 0.20377661$

Step 3.3.2.1.6

Multiply $18$ by $0.20377661$ .

$f (0.45141622) = 0.12457472 - 1.47180912 + 3.66797902$

Step 3.3.2.2

Simplify by adding and subtracting.

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Step 3.3.2.2.1

Subtract $1.47180912$ from $0.12457472$ .

$f (0.45141622) = - 1.34723439 + 3.66797902$

Step 3.3.2.2.2

Add $- 1.34723439$ and $3.66797902$ .

$f (0.45141622) = 2.32074462$

Step 3.3.2.3

The final answer is $2.32074462$ .

$2.32074462$

Step 3.4

The point found by substituting $0.45141622$ in $f (x) = 3 x^{4} - 16 x^{3} + 18 x^{2}$ is $(0.45141622, 2.32074462)$ . This point can be an inflection point.

$(0.45141622, 2.32074462)$

Step 3.5

Determine the points that could be inflection points.

$(2.21525043, - 13.35778166), (0.45141622, 2.32074462)$

Step 4

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

$(- \infty, 0.45141622) \cup (0.45141622, 2.21525043) \cup (2.21525043, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 0.45141622)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $0.35141622$ in the expression.

$f'' (0.35141622) = 36 {(0.35141622)}^{2} - 96 \cdot 0.35141622 + 36$

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

Raise $0.35141622$ to the power of $2$ .

$f'' (0.35141622) = 36 \cdot 0.12349336 - 96 \cdot 0.35141622 + 36$

Step 5.2.1.2

Multiply $36$ by $0.12349336$ .

$f'' (0.35141622) = 4.44576119 - 96 \cdot 0.35141622 + 36$

Step 5.2.1.3

Multiply $- 96$ by $0.35141622$ .

$f'' (0.35141622) = 4.44576119 - 33.73595804 + 36$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $33.73595804$ from $4.44576119$ .

$f'' (0.35141622) = - 29.29019685 + 36$

Step 5.2.2.2

Add $- 29.29019685$ and $36$ .

$f'' (0.35141622) = 6.70980314$

Step 5.2.3

The final answer is $6.70980314$ .

$6.70980314$

Step 5.3

At $0.35141622$ , the second derivative is $6.70980314$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 0.45141622)$ .

Increasing on $(- \infty, 0.45141622)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(0.45141622, 2.21525043)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $1.\overline{3}$ in the expression.

$f'' (1.\overline{3}) = 36 {(1.\overline{3})}^{2} - 96 \cdot 1.\overline{3} + 36$

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 $1.\overline{3}$ to the power of $2$ .

$f'' (1.\overline{3}) = 36 \cdot 1.\overline{7} - 96 \cdot 1.\overline{3} + 36$

Step 6.2.1.2

Multiply $36$ by $1.\overline{7}$ .

$f'' (1.\overline{3}) = 64 - 96 \cdot 1.\overline{3} + 36$

Step 6.2.1.3

Multiply $- 96$ by $1.\overline{3}$ .

$f'' (1.\overline{3}) = 64 - 128 + 36$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $128$ from $64$ .

$f'' (1.\overline{3}) = - 64 + 36$

Step 6.2.2.2

Add $- 64$ and $36$ .

$f'' (1.\overline{3}) = - 28$

Step 6.2.3

The final answer is $- 28$ .

$- 28$

Step 6.3

At $1.\overline{3}$ , the second derivative is $- 28$ . Since this is negative, the second derivative is decreasing on the interval $(0.45141622, 2.21525043)$

Decreasing on $(0.45141622, 2.21525043)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(2.21525043, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $2.31525043$ in the expression.

$f'' (2.31525043) = 36 {(2.31525043)}^{2} - 96 \cdot 2.31525043 + 36$

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 $2.31525043$ to the power of $2$ .

$f'' (2.31525043) = 36 \cdot 5.36038458 - 96 \cdot 2.31525043 + 36$

Step 7.2.1.2

Multiply $36$ by $5.36038458$ .

$f'' (2.31525043) = 192.9738451 - 96 \cdot 2.31525043 + 36$

Step 7.2.1.3

Multiply $- 96$ by $2.31525043$ .

$f'' (2.31525043) = 192.9738451 - 222.26404195 + 36$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $222.26404195$ from $192.9738451$ .

$f'' (2.31525043) = - 29.29019685 + 36$

Step 7.2.2.2

Add $- 29.29019685$ and $36$ .

$f'' (2.31525043) = 6.70980314$

Step 7.2.3

The final answer is $6.70980314$ .

$6.70980314$

Step 7.3

At $2.31525043$ , the second derivative is $6.70980314$ . Since this is positive, the second derivative is increasing on the interval $(2.21525043, \infty)$ .

Increasing on $(2.21525043, \infty)$ since $f'' (x) > 0$

Step 8

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 $(0.45141622, 2.32074462), (2.21525043, - 13.35778166)$ .

$(0.45141622, 2.32074462), (2.21525043, - 13.35778166)$

Step 9