Calculus Examples

$f (x) = x^{4} + 8 x^{3} + 18 x^{2} + 8$

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 $x^{4} + 8 x^{3} + 18 x^{2} + 8$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [18 x^{2}] + \frac{d}{d x} [8]$ .

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (8 x^{3}) + \frac{d}{d x} (18 x^{2}) + \frac{d}{d x} (8)$

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 $n = 4$ .

$f' (x) = 4 x^{3} + \frac{d}{d x} (8 x^{3}) + \frac{d}{d x} (18 x^{2}) + \frac{d}{d x} (8)$

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

$f' (x) = 4 x^{3} + 8 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (18 x^{2}) + \frac{d}{d x} (8)$

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 $n = 3$ .

$f' (x) = 4 x^{3} + 8 (3 x^{2}) + \frac{d}{d x} (18 x^{2}) + \frac{d}{d x} (8)$

Step 1.1.2.3

Multiply $3$ by $8$ .

$f' (x) = 4 x^{3} + 24 x^{2} + \frac{d}{d x} (18 x^{2}) + \frac{d}{d x} (8)$

Step 1.1.3

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

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Step 1.1.3.1

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

$f' (x) = 4 x^{3} + 24 x^{2} + 18 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (8)$

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 $n = 2$ .

$f' (x) = 4 x^{3} + 24 x^{2} + 18 (2 x) + \frac{d}{d x} (8)$

Step 1.1.3.3

Multiply $2$ by $18$ .

$f' (x) = 4 x^{3} + 24 x^{2} + 36 x + \frac{d}{d x} (8)$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f' (x) = 4 x^{3} + 24 x^{2} + 36 x + 0$

Step 1.1.4.2

Add $4 x^{3} + 24 x^{2} + 36 x$ and $0$ .

$f' (x) = 4 x^{3} + 24 x^{2} + 36 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $4 x^{3} + 24 x^{2} + 36 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [36 x]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (24 x^{2}) + \frac{d}{d x} (36 x)$

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

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (24 x^{2}) + \frac{d}{d x} (36 x)$

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 $n = 3$ .

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (24 x^{2}) + \frac{d}{d x} (36 x)$

Step 1.2.2.3

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (24 x^{2}) + \frac{d}{d x} (36 x)$

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

$f'' (x) = 12 x^{2} + 24 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (36 x)$

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 $n = 2$ .

$f'' (x) = 12 x^{2} + 24 (2 x) + \frac{d}{d x} (36 x)$

Step 1.2.3.3

Multiply $2$ by $24$ .

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

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

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

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 $n = 1$ .

$f'' (x) = 12 x^{2} + 48 x + 36 \cdot 1$

Step 1.2.4.3

Multiply $36$ by $1$ .

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

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $12 x^{2} + 48 x + 36$ .

$12 x^{2} + 48 x + 36$

Step 2

Set the second derivative equal to $0$ then solve the equation $12 x^{2} + 48 x + 36 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$12 x^{2} + 48 x + 36 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $12$ out of $12 x^{2} + 48 x + 36$ .

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Step 2.2.1.1

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

$12 (x^{2}) + 48 x + 36 = 0$

Step 2.2.1.2

Factor $12$ out of $48 x$ .

$12 (x^{2}) + 12 (4 x) + 36 = 0$

Step 2.2.1.3

Factor $12$ out of $36$ .

$12 x^{2} + 12 (4 x) + 12 \cdot 3 = 0$

Step 2.2.1.4

Factor $12$ out of $12 x^{2} + 12 (4 x)$ .

$12 (x^{2} + 4 x) + 12 \cdot 3 = 0$

Step 2.2.1.5

Factor $12$ out of $12 (x^{2} + 4 x) + 12 \cdot 3$ .

$12 (x^{2} + 4 x + 3) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $x^{2} + 4 x + 3$ using the AC method.

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Step 2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Step 2.2.2.1.2

Write the factored form using these integers.

$12 ((x + 1) (x + 3)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$12 (x + 1) (x + 3) = 0$

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 $0$ .

$x + 1 = 0$

$x + 3 = 0$

Step 2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

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

$x + 3 = 0$

Step 2.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.6

The final solution is all the values that make $12 (x + 1) (x + 3) = 0$ true.

$x = - 1, - 3$

Step 3

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

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Step 3.1

Substitute $- 1$ in $f (x) = x^{4} + 8 x^{3} + 18 x^{2} + 8$ to find the value of $y$ .

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Step 3.1.1

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

$f (- 1) = {(- 1)}^{4} + 8 {(- 1)}^{3} + 18 {(- 1)}^{2} + 8$

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 $- 1$ to the power of $4$ .

$f (- 1) = 1 + 8 {(- 1)}^{3} + 18 {(- 1)}^{2} + 8$

Step 3.1.2.1.2

Raise $- 1$ to the power of $3$ .

$f (- 1) = 1 + 8 \cdot - 1 + 18 {(- 1)}^{2} + 8$

Step 3.1.2.1.3

Multiply $8$ by $- 1$ .

$f (- 1) = 1 - 8 + 18 {(- 1)}^{2} + 8$

Step 3.1.2.1.4

Raise $- 1$ to the power of $2$ .

$f (- 1) = 1 - 8 + 18 \cdot 1 + 8$

Step 3.1.2.1.5

Multiply $18$ by $1$ .

$f (- 1) = 1 - 8 + 18 + 8$

Step 3.1.2.2

Simplify by adding and subtracting.

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Step 3.1.2.2.1

Subtract $8$ from $1$ .

$f (- 1) = - 7 + 18 + 8$

Step 3.1.2.2.2

Add $- 7$ and $18$ .

$f (- 1) = 11 + 8$

Step 3.1.2.2.3

Add $11$ and $8$ .

$f (- 1) = 19$

Step 3.1.2.3

The final answer is $19$ .

$19$

Step 3.2

The point found by substituting $- 1$ in $f (x) = x^{4} + 8 x^{3} + 18 x^{2} + 8$ is $(- 1, 19)$ . This point can be an inflection point.

$(- 1, 19)$

Step 3.3

Substitute $- 3$ in $f (x) = x^{4} + 8 x^{3} + 18 x^{2} + 8$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = {(- 3)}^{4} + 8 {(- 3)}^{3} + 18 {(- 3)}^{2} + 8$

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

$f (- 3) = 81 + 8 {(- 3)}^{3} + 18 {(- 3)}^{2} + 8$

Step 3.3.2.1.2

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

$f (- 3) = 81 + 8 \cdot - 27 + 18 {(- 3)}^{2} + 8$

Step 3.3.2.1.3

Multiply $8$ by $- 27$ .

$f (- 3) = 81 - 216 + 18 {(- 3)}^{2} + 8$

Step 3.3.2.1.4

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

$f (- 3) = 81 - 216 + 18 \cdot 9 + 8$

Step 3.3.2.1.5

Multiply $18$ by $9$ .

$f (- 3) = 81 - 216 + 162 + 8$

Step 3.3.2.2

Simplify by adding and subtracting.

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Step 3.3.2.2.1

Subtract $216$ from $81$ .

$f (- 3) = - 135 + 162 + 8$

Step 3.3.2.2.2

Add $- 135$ and $162$ .

$f (- 3) = 27 + 8$

Step 3.3.2.2.3

Add $27$ and $8$ .

$f (- 3) = 35$

Step 3.3.2.3

The final answer is $35$ .

$35$

Step 3.4

The point found by substituting $- 3$ in $f (x) = x^{4} + 8 x^{3} + 18 x^{2} + 8$ is $(- 3, 35)$ . This point can be an inflection point.

$(- 3, 35)$

Step 3.5

Determine the points that could be inflection points.

$(- 1, 19), (- 3, 35)$

Step 4

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

$(- \infty, - 3) \cup (- 3, - 1) \cup (- 1, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 3)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 3.1$ in the expression.

$f'' (- 3.1) = 12 {(- 3.1)}^{2} + 48 (- 3.1) + 36$

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

Raise $- 3.1$ to the power of $2$ .

$f'' (- 3.1) = 12 \cdot 9.61 + 48 (- 3.1) + 36$

Step 5.2.1.2

Multiply $12$ by $9.61$ .

$f'' (- 3.1) = 115.32 + 48 (- 3.1) + 36$

Step 5.2.1.3

Multiply $48$ by $- 3.1$ .

$f'' (- 3.1) = 115.32 - 148.8 + 36$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $148.8$ from $115.32$ .

$f'' (- 3.1) = - 33.48 + 36$

Step 5.2.2.2

Add $- 33.48$ and $36$ .

$f'' (- 3.1) = 2.52$

Step 5.2.3

The final answer is $2.52$ .

$2.52$

Step 5.3

At $- 3.1$ , the second derivative is $2.52$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 3)$ .

Increasing on $(- \infty, - 3)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- 3, - 1)$ 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.

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

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

$f'' (- 2) = 12 \cdot 4 + 48 (- 2) + 36$

Step 6.2.1.2

Multiply $12$ by $4$ .

$f'' (- 2) = 48 + 48 (- 2) + 36$

Step 6.2.1.3

Multiply $48$ by $- 2$ .

$f'' (- 2) = 48 - 96 + 36$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $96$ from $48$ .

$f'' (- 2) = - 48 + 36$

Step 6.2.2.2

Add $- 48$ and $36$ .

$f'' (- 2) = - 12$

Step 6.2.3

The final answer is $- 12$ .

$- 12$

Step 6.3

At $- 2$ , the second derivative is $- 12$ . Since this is negative, the second derivative is decreasing on the interval $(- 3, - 1)$

Decreasing on $(- 3, - 1)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 1, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 0.9$ in the expression.

$f'' (- 0.9) = 12 {(- 0.9)}^{2} + 48 (- 0.9) + 36$

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 $- 0.9$ to the power of $2$ .

$f'' (- 0.9) = 12 \cdot 0.81 + 48 (- 0.9) + 36$

Step 7.2.1.2

Multiply $12$ by $0.81$ .

$f'' (- 0.9) = 9.72 + 48 (- 0.9) + 36$

Step 7.2.1.3

Multiply $48$ by $- 0.9$ .

$f'' (- 0.9) = 9.72 - 43.2 + 36$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $43.2$ from $9.72$ .

$f'' (- 0.9) = - 33.48 + 36$

Step 7.2.2.2

Add $- 33.48$ and $36$ .

$f'' (- 0.9) = 2.52$

Step 7.2.3

The final answer is $2.52$ .

$2.52$

Step 7.3

At $- 0.9$ , the second derivative is $2.52$ . Since this is positive, the second derivative is increasing on the interval $(- 1, \infty)$ .

Increasing on $(- 1, \infty)$ 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 $(- 3, 35), (- 1, 19)$ .

$(- 3, 35), (- 1, 19)$

Step 9