Calculus Examples

$f (x) = x^{4} + 8 x^{3}$

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

Differentiate.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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 $n = 3$ .

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

Step 1.1.2.3

Multiply $3$ by $8$ .

$f' (x) = 4 x^{3} + 24 x^{2}$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $4 x^{3} + 24 x^{2}$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [24 x^{2}]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (24 x^{2})$

Step 1.2.2

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

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Step 1.2.2.1

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

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (24 x^{2})$

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 $n = 3$ .

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (24 x^{2})$

Step 1.2.2.3

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (24 x^{2})$

Step 1.2.3

Evaluate $\frac{d}{d x} [24 x^{2}]$ .

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Step 1.2.3.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x^{2}$ with respect to $x$ is $24 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 x^{2} + 24 \frac{d}{d x} (x^{2})$

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 $n = 2$ .

$f'' (x) = 12 x^{2} + 24 (2 x)$

Step 1.2.3.3

Multiply $2$ by $24$ .

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

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $12 x^{2} + 48 x$ .

$12 x^{2} + 48 x$

Step 2

Set the second derivative equal to $0$ then solve the equation $12 x^{2} + 48 x = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$12 x^{2} + 48 x = 0$

Step 2.2

Factor $12 x$ out of $12 x^{2} + 48 x$ .

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Step 2.2.1

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

$12 x (x) + 48 x = 0$

Step 2.2.2

Factor $12 x$ out of $48 x$ .

$12 x (x) + 12 x (4) = 0$

Step 2.2.3

Factor $12 x$ out of $12 x (x) + 12 x (4)$ .

$12 x (x + 4) = 0$

Step 2.3

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

$x = 0$

$x + 4 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.6

The final solution is all the values that make $12 x (x + 4) = 0$ true.

$x = 0, - 4$

Step 3

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

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Step 3.1

Substitute $0$ in $f (x) = x^{4} + 8 x^{3}$ to find the value of $y$ .

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Step 3.1.1

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

$f (0) = {(0)}^{4} + 8 {(0)}^{3}$

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

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

$f (0) = 0 + 8 {(0)}^{3}$

Step 3.1.2.1.2

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

$f (0) = 0 + 8 \cdot 0$

Step 3.1.2.1.3

Multiply $8$ by $0$ .

$f (0) = 0 + 0$

Step 3.1.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 3.1.2.3

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $0$ in $f (x) = x^{4} + 8 x^{3}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.3

Substitute $- 4$ in $f (x) = x^{4} + 8 x^{3}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = {(- 4)}^{4} + 8 {(- 4)}^{3}$

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

$f (- 4) = 256 + 8 {(- 4)}^{3}$

Step 3.3.2.1.2

Raise $- 4$ to the power of $3$ .

$f (- 4) = 256 + 8 \cdot - 64$

Step 3.3.2.1.3

Multiply $8$ by $- 64$ .

$f (- 4) = 256 - 512$

Step 3.3.2.2

Subtract $512$ from $256$ .

$f (- 4) = - 256$

Step 3.3.2.3

The final answer is $- 256$ .

$- 256$

Step 3.4

The point found by substituting $- 4$ in $f (x) = x^{4} + 8 x^{3}$ is $(- 4, - 256)$ . This point can be an inflection point.

$(- 4, - 256)$

Step 3.5

Determine the points that could be inflection points.

$(0, 0), (- 4, - 256)$

Step 4

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

$(- \infty, - 4) \cup (- 4, 0) \cup (0, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 4)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 4.1$ in the expression.

$f'' (- 4.1) = 12 {(- 4.1)}^{2} + 48 (- 4.1)$

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 $- 4.1$ to the power of $2$ .

$f'' (- 4.1) = 12 \cdot 16.81 + 48 (- 4.1)$

Step 5.2.1.2

Multiply $12$ by $16.81$ .

$f'' (- 4.1) = 201.72 + 48 (- 4.1)$

Step 5.2.1.3

Multiply $48$ by $- 4.1$ .

$f'' (- 4.1) = 201.72 - 196.8$

Step 5.2.2

Subtract $196.8$ from $201.72$ .

$f'' (- 4.1) = 4.92$

Step 5.2.3

The final answer is $4.92$ .

$4.92$

Step 5.3

At $- 4.1$ , the second derivative is $4.92$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 4)$ .

Increasing on $(- \infty, - 4)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- 4, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

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

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

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 $2$ .

$f'' (- 2) = 12 \cdot 4 + 48 (- 2)$

Step 6.2.1.2

Multiply $12$ by $4$ .

$f'' (- 2) = 48 + 48 (- 2)$

Step 6.2.1.3

Multiply $48$ by $- 2$ .

$f'' (- 2) = 48 - 96$

Step 6.2.2

Subtract $96$ from $48$ .

$f'' (- 2) = - 48$

Step 6.2.3

The final answer is $- 48$ .

$- 48$

Step 6.3

At $- 2$ , the second derivative is $- 48$ . Since this is negative, the second derivative is decreasing on the interval $(- 4, 0)$

Decreasing on $(- 4, 0)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(0, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.1$ in the expression.

$f'' (0.1) = 12 {(0.1)}^{2} + 48 (0.1)$

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 $0.1$ to the power of $2$ .

$f'' (0.1) = 12 \cdot 0.01 + 48 (0.1)$

Step 7.2.1.2

Multiply $12$ by $0.01$ .

$f'' (0.1) = 0.12 + 48 (0.1)$

Step 7.2.1.3

Multiply $48$ by $0.1$ .

$f'' (0.1) = 0.12 + 4.8$

Step 7.2.2

Add $0.12$ and $4.8$ .

$f'' (0.1) = 4.92$

Step 7.2.3

The final answer is $4.92$ .

$4.92$

Step 7.3

At $0.1$ , the second derivative is $4.92$ . Since this is positive, the second derivative is increasing on the interval $(0, \infty)$ .

Increasing on $(0, \infty)$ 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 $(- 4, - 256), (0, 0)$ .

$(- 4, - 256), (0, 0)$

Step 9