Calculus Examples

$5 x^{4} + 20 x^{3}$

Step 1

Write $5 x^{4} + 20 x^{3}$ as a function.

$f (x) = 5 x^{4} + 20 x^{3}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $5 x^{4} + 20 x^{3}$ with respect to $x$ is $\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [20 x^{3}]$ .

$\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [20 x^{3}]$

Step 2.1.2

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

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Step 2.1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [20 x^{3}]$

Step 2.1.2.2

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

$5 (4 x^{3}) + \frac{d}{d x} [20 x^{3}]$

Step 2.1.2.3

Multiply $4$ by $5$ .

$20 x^{3} + \frac{d}{d x} [20 x^{3}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [20 x^{3}]$ .

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Step 2.1.3.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x^{3}$ with respect to $x$ is $20 \frac{d}{d x} [x^{3}]$ .

$20 x^{3} + 20 \frac{d}{d x} [x^{3}]$

Step 2.1.3.2

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

$20 x^{3} + 20 (3 x^{2})$

Step 2.1.3.3

Multiply $3$ by $20$ .

$f' (x) = 20 x^{3} + 60 x^{2}$

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $20 x^{3} + 60 x^{2}$ with respect to $x$ is $\frac{d}{d x} [20 x^{3}] + \frac{d}{d x} [60 x^{2}]$ .

$\frac{d}{d x} [20 x^{3}] + \frac{d}{d x} [60 x^{2}]$

Step 2.2.2

Evaluate $\frac{d}{d x} [20 x^{3}]$ .

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Step 2.2.2.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x^{3}$ with respect to $x$ is $20 \frac{d}{d x} [x^{3}]$ .

$20 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [60 x^{2}]$

Step 2.2.2.2

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

$20 (3 x^{2}) + \frac{d}{d x} [60 x^{2}]$

Step 2.2.2.3

Multiply $3$ by $20$ .

$60 x^{2} + \frac{d}{d x} [60 x^{2}]$

Step 2.2.3

Evaluate $\frac{d}{d x} [60 x^{2}]$ .

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Step 2.2.3.1

Since $60$ is constant with respect to $x$ , the derivative of $60 x^{2}$ with respect to $x$ is $60 \frac{d}{d x} [x^{2}]$ .

$60 x^{2} + 60 \frac{d}{d x} [x^{2}]$

Step 2.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$60 x^{2} + 60 (2 x)$

Step 2.2.3.3

Multiply $2$ by $60$ .

$f'' (x) = 60 x^{2} + 120 x$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $60 x^{2} + 120 x$ .

$60 x^{2} + 120 x$

Step 3

Set the second derivative equal to $0$ then solve the equation $60 x^{2} + 120 x = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$60 x^{2} + 120 x = 0$

Step 3.2

Factor $60 x$ out of $60 x^{2} + 120 x$ .

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Step 3.2.1

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

$60 x (x) + 120 x = 0$

Step 3.2.2

Factor $60 x$ out of $120 x$ .

$60 x (x) + 60 x (2) = 0$

Step 3.2.3

Factor $60 x$ out of $60 x (x) + 60 x (2)$ .

$60 x (x + 2) = 0$

Step 3.3

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

$x = 0$

$x + 2 = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.6

The final solution is all the values that make $60 x (x + 2) = 0$ true.

$x = 0, - 2$

Step 4

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

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Step 4.1

Substitute $0$ in $f (x) = 5 x^{4} + 20 x^{3}$ to find the value of $y$ .

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Step 4.1.1

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

$f (0) = 5 {(0)}^{4} + 20 {(0)}^{3}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$f (0) = 5 \cdot 0 + 20 {(0)}^{3}$

Step 4.1.2.1.2

Multiply $5$ by $0$ .

$f (0) = 0 + 20 {(0)}^{3}$

Step 4.1.2.1.3

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

$f (0) = 0 + 20 \cdot 0$

Step 4.1.2.1.4

Multiply $20$ by $0$ .

$f (0) = 0 + 0$

Step 4.1.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 4.1.2.3

The final answer is $0$ .

$0$

Step 4.2

The point found by substituting $0$ in $f (x) = 5 x^{4} + 20 x^{3}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4.3

Substitute $- 2$ in $f (x) = 5 x^{4} + 20 x^{3}$ to find the value of $y$ .

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Step 4.3.1

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

$f (- 2) = 5 {(- 2)}^{4} + 20 {(- 2)}^{3}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

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

$f (- 2) = 5 \cdot 16 + 20 {(- 2)}^{3}$

Step 4.3.2.1.2

Multiply $5$ by $16$ .

$f (- 2) = 80 + 20 {(- 2)}^{3}$

Step 4.3.2.1.3

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

$f (- 2) = 80 + 20 \cdot - 8$

Step 4.3.2.1.4

Multiply $20$ by $- 8$ .

$f (- 2) = 80 - 160$

Step 4.3.2.2

Subtract $160$ from $80$ .

$f (- 2) = - 80$

Step 4.3.2.3

The final answer is $- 80$ .

$- 80$

Step 4.4

The point found by substituting $- 2$ in $f (x) = 5 x^{4} + 20 x^{3}$ is $(- 2, - 80)$ . This point can be an inflection point.

$(- 2, - 80)$

Step 4.5

Determine the points that could be inflection points.

$(0, 0), (- 2, - 80)$

Step 5

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

$(- \infty, - 2) \cup (- 2, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 2)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 2.1$ in the expression.

$f'' (- 2.1) = 60 {(- 2.1)}^{2} + 120 (- 2.1)$

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 $- 2.1$ to the power of $2$ .

$f'' (- 2.1) = 60 \cdot 4.41 + 120 (- 2.1)$

Step 6.2.1.2

Multiply $60$ by $4.41$ .

$f'' (- 2.1) = 264.6 + 120 (- 2.1)$

Step 6.2.1.3

Multiply $120$ by $- 2.1$ .

$f'' (- 2.1) = 264.6 - 252$

Step 6.2.2

Subtract $252$ from $264.6$ .

$f'' (- 2.1) = 12.6$

Step 6.2.3

The final answer is $12.6$ .

$12.6$

Step 6.3

At $- 2.1$ , the second derivative is $12.6$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 2)$ .

Increasing on $(- \infty, - 2)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(- 2, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

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

$f'' (- 1) = 60 {(- 1)}^{2} + 120 (- 1)$

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

$f'' (- 1) = 60 \cdot 1 + 120 (- 1)$

Step 7.2.1.2

Multiply $60$ by $1$ .

$f'' (- 1) = 60 + 120 (- 1)$

Step 7.2.1.3

Multiply $120$ by $- 1$ .

$f'' (- 1) = 60 - 120$

Step 7.2.2

Subtract $120$ from $60$ .

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

Step 7.2.3

The final answer is $- 60$ .

$- 60$

Step 7.3

At $- 1$ , the second derivative is $- 60$ . Since this is negative, the second derivative is decreasing on the interval $(- 2, 0)$

Decreasing on $(- 2, 0)$ since $f'' (x) < 0$

Step 8

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 8.1

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

$f'' (0.1) = 60 {(0.1)}^{2} + 120 (0.1)$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $0.1$ to the power of $2$ .

$f'' (0.1) = 60 \cdot 0.01 + 120 (0.1)$

Step 8.2.1.2

Multiply $60$ by $0.01$ .

$f'' (0.1) = 0.6 + 120 (0.1)$

Step 8.2.1.3

Multiply $120$ by $0.1$ .

$f'' (0.1) = 0.6 + 12$

Step 8.2.2

Add $0.6$ and $12$ .

$f'' (0.1) = 12.6$

Step 8.2.3

The final answer is $12.6$ .

$12.6$

Step 8.3

At $0.1$ , the second derivative is $12.6$ . Since this is positive, the second derivative is increasing on the interval $(0, \infty)$ .

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

Step 9

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 $(- 2, - 80), (0, 0)$ .

$(- 2, - 80), (0, 0)$

Step 10