Calculus Examples

$\frac{3}{20} x^{5} - 2 x^{4} + 8 x^{3}$

Step 1

Write $\frac{3}{20} x^{5} - 2 x^{4} + 8 x^{3}$ as a function.

$f (x) = \frac{3}{20} x^{5} - 2 x^{4} + 8 x^{3}$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

By the Sum Rule, the derivative of $\frac{3}{20} x^{5} - 2 x^{4} + 8 x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{20} x^{5}] + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}]$ .

$\frac{d}{d x} [\frac{3}{20} x^{5}] + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.2

Evaluate $\frac{d}{d x} [\frac{3}{20} x^{5}]$ .

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Step 2.1.1.2.1

Since $\frac{3}{20}$ is constant with respect to $x$ , the derivative of $\frac{3}{20} x^{5}$ with respect to $x$ is $\frac{3}{20} \frac{d}{d x} [x^{5}]$ .

$\frac{3}{20} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{3}{20} (5 x^{4}) + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.2.3

Combine $5$ and $\frac{3}{20}$ .

$\frac{5 \cdot 3}{20} x^{4} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.2.4

Multiply $5$ by $3$ .

$\frac{15}{20} x^{4} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.2.5

Combine $\frac{15}{20}$ and $x^{4}$ .

$\frac{15 x^{4}}{20} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.2.6

Cancel the common factor of $15$ and $20$ .

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Step 2.1.1.2.6.1

Factor $5$ out of $15 x^{4}$ .

$\frac{5 (3 x^{4})}{20} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.2.6.2

Cancel the common factors.

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Step 2.1.1.2.6.2.1

Factor $5$ out of $20$ .

$\frac{5 (3 x^{4})}{5 (4)} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.2.6.2.2

Cancel the common factor.

$\frac{5 (3 x^{4})}{5 \cdot 4} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.2.6.2.3

Rewrite the expression.

$\frac{3 x^{4}}{4} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.3

Evaluate $\frac{d}{d x} [- 2 x^{4}]$ .

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Step 2.1.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{4}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{4}]$ .

$\frac{3 x^{4}}{4} - 2 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.3.2

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

$\frac{3 x^{4}}{4} - 2 (4 x^{3}) + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.3.3

Multiply $4$ by $- 2$ .

$\frac{3 x^{4}}{4} - 8 x^{3} + \frac{d}{d x} [8 x^{3}]$

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

$\frac{3 x^{4}}{4} - 8 x^{3} + 8 \frac{d}{d x} [x^{3}]$

Step 2.1.1.4.2

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

$\frac{3 x^{4}}{4} - 8 x^{3} + 8 (3 x^{2})$

Step 2.1.1.4.3

Multiply $3$ by $8$ .

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

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $\frac{3 x^{4}}{4} - 8 x^{3} + 24 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{3 x^{4}}{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$ .

$\frac{d}{d x} [\frac{3 x^{4}}{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$\frac{3}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.2.2

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

$\frac{3}{4} (4 x^{3}) + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.2.3

Combine $4$ and $\frac{3}{4}$ .

$\frac{4 \cdot 3}{4} x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.2.4

Multiply $4$ by $3$ .

$\frac{12}{4} x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.2.5

Combine $\frac{12}{4}$ and $x^{3}$ .

$\frac{12 x^{3}}{4} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.2.6

Cancel the common factor of $12$ and $4$ .

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Step 2.1.2.2.6.1

Factor $4$ out of $12 x^{3}$ .

$\frac{4 (3 x^{3})}{4} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.2.6.2

Cancel the common factors.

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Step 2.1.2.2.6.2.1

Factor $4$ out of $4$ .

$\frac{4 (3 x^{3})}{4 (1)} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.2.6.2.2

Cancel the common factor.

$\frac{4 (3 x^{3})}{4 \cdot 1} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.2.6.2.3

Rewrite the expression.

$\frac{3 x^{3}}{1} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.2.6.2.4

Divide $3 x^{3}$ by $1$ .

$3 x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.3

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

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Step 2.1.2.3.1

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

$3 x^{3} - 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.3.2

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

$3 x^{3} - 8 (3 x^{2}) + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.3.3

Multiply $3$ by $- 8$ .

$3 x^{3} - 24 x^{2} + \frac{d}{d x} [24 x^{2}]$

Step 2.1.2.4

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

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Step 2.1.2.4.1

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

$3 x^{3} - 24 x^{2} + 24 \frac{d}{d x} [x^{2}]$

Step 2.1.2.4.2

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

$3 x^{3} - 24 x^{2} + 24 (2 x)$

Step 2.1.2.4.3

Multiply $2$ by $24$ .

$f'' (x) = 3 x^{3} - 24 x^{2} + 48 x$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $3 x^{3} - 24 x^{2} + 48 x$ .

$3 x^{3} - 24 x^{2} + 48 x$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $3 x^{3} - 24 x^{2} + 48 x = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$3 x^{3} - 24 x^{2} + 48 x = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $3 x$ out of $3 x^{3} - 24 x^{2} + 48 x$ .

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Step 2.2.2.1.1

Factor $3 x$ out of $3 x^{3}$ .

$3 x (x^{2}) - 24 x^{2} + 48 x = 0$

Step 2.2.2.1.2

Factor $3 x$ out of $- 24 x^{2}$ .

$3 x (x^{2}) + 3 x (- 8 x) + 48 x = 0$

Step 2.2.2.1.3

Factor $3 x$ out of $48 x$ .

$3 x (x^{2}) + 3 x (- 8 x) + 3 x (16) = 0$

Step 2.2.2.1.4

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

$3 x (x^{2} - 8 x) + 3 x (16) = 0$

Step 2.2.2.1.5

Factor $3 x$ out of $3 x (x^{2} - 8 x) + 3 x (16)$ .

$3 x (x^{2} - 8 x + 16) = 0$

Step 2.2.2.2

Factor using the perfect square rule.

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Step 2.2.2.2.1

Rewrite $16$ as $4^{2}$ .

$3 x (x^{2} - 8 x + 4^{2}) = 0$

Step 2.2.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 2.2.2.2.3

Rewrite the polynomial.

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

Step 2.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Step 2.2.3

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

$x = 0$

${(x - 4)}^{2} = 0$

Step 2.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.5

Set ${(x - 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set ${(x - 4)}^{2}$ equal to $0$ .

${(x - 4)}^{2} = 0$

Step 2.2.5.2

Solve ${(x - 4)}^{2} = 0$ for $x$ .

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Step 2.2.5.2.1

Set the $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.2.5.2.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.2.6

The final solution is all the values that make $3 x {(x - 4)}^{2} = 0$ true.

$x = 0, 4$

Step 3

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 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

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

Step 5

Substitute any number from the interval $(- \infty, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (- 2) = 3 {(- 2)}^{3} - 24 {(- 2)}^{2} + 48 (- 2)$

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

$f'' (- 2) = 3 \cdot - 8 - 24 {(- 2)}^{2} + 48 (- 2)$

Step 5.2.1.2

Multiply $3$ by $- 8$ .

$f'' (- 2) = - 24 - 24 {(- 2)}^{2} + 48 (- 2)$

Step 5.2.1.3

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

$f'' (- 2) = - 24 - 24 \cdot 4 + 48 (- 2)$

Step 5.2.1.4

Multiply $- 24$ by $4$ .

$f'' (- 2) = - 24 - 96 + 48 (- 2)$

Step 5.2.1.5

Multiply $48$ by $- 2$ .

$f'' (- 2) = - 24 - 96 - 96$

Step 5.2.2

Simplify by subtracting numbers.

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Step 5.2.2.1

Subtract $96$ from $- 24$ .

$f'' (- 2) = - 120 - 96$

Step 5.2.2.2

Subtract $96$ from $- 120$ .

$f'' (- 2) = - 216$

Step 5.2.3

The final answer is $- 216$ .

$- 216$

Step 5.3

The graph is concave down on the interval $(- \infty, 0)$ because $f'' (- 2)$ is negative.

Concave down on $(- \infty, 0)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(0, 4)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (2) = 3 {(2)}^{3} - 24 {(2)}^{2} + 48 (2)$

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 $3$ .

$f'' (2) = 3 \cdot 8 - 24 {(2)}^{2} + 48 (2)$

Step 6.2.1.2

Multiply $3$ by $8$ .

$f'' (2) = 24 - 24 {(2)}^{2} + 48 (2)$

Step 6.2.1.3

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

$f'' (2) = 24 - 24 \cdot 4 + 48 (2)$

Step 6.2.1.4

Multiply $- 24$ by $4$ .

$f'' (2) = 24 - 96 + 48 (2)$

Step 6.2.1.5

Multiply $48$ by $2$ .

$f'' (2) = 24 - 96 + 96$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $96$ from $24$ .

$f'' (2) = - 72 + 96$

Step 6.2.2.2

Add $- 72$ and $96$ .

$f'' (2) = 24$

Step 6.2.3

The final answer is $24$ .

$24$

Step 6.3

The graph is concave up on the interval $(0, 4)$ because $f'' (2)$ is positive.

Concave up on $(0, 4)$ since $f'' (x)$ is positive

Step 7

Substitute any number from the interval $(4, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $6$ in the expression.

$f'' (6) = 3 {(6)}^{3} - 24 {(6)}^{2} + 48 (6)$

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 $6$ to the power of $3$ .

$f'' (6) = 3 \cdot 216 - 24 {(6)}^{2} + 48 (6)$

Step 7.2.1.2

Multiply $3$ by $216$ .

$f'' (6) = 648 - 24 {(6)}^{2} + 48 (6)$

Step 7.2.1.3

Raise $6$ to the power of $2$ .

$f'' (6) = 648 - 24 \cdot 36 + 48 (6)$

Step 7.2.1.4

Multiply $- 24$ by $36$ .

$f'' (6) = 648 - 864 + 48 (6)$

Step 7.2.1.5

Multiply $48$ by $6$ .

$f'' (6) = 648 - 864 + 288$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $864$ from $648$ .

$f'' (6) = - 216 + 288$

Step 7.2.2.2

Add $- 216$ and $288$ .

$f'' (6) = 72$

Step 7.2.3

The final answer is $72$ .

$72$

Step 7.3

The graph is concave up on the interval $(4, \infty)$ because $f'' (6)$ is positive.

Concave up on $(4, \infty)$ since $f'' (x)$ is positive

Step 8

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on $(- \infty, 0)$ since $f'' (x)$ is negative

Concave up on $(0, 4)$ since $f'' (x)$ is positive

Concave up on $(4, \infty)$ since $f'' (x)$ is positive

Step 9