Calculus Examples

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

Step 1

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

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

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 $\frac{3}{20} x^{5} - 2 x^{4}$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{20} x^{5}] + \frac{d}{d x} [- 2 x^{4}]$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

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 $n = 5$ .

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

Step 2.1.2.3

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

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

Step 2.1.2.4

Multiply $5$ by $3$ .

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

Step 2.1.2.5

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

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

Step 2.1.2.6

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

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Step 2.1.2.6.1

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

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

Step 2.1.2.6.2

Cancel the common factors.

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Step 2.1.2.6.2.1

Factor $5$ out of $20$ .

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

Step 2.1.2.6.2.2

Cancel the common factor.

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

Step 2.1.2.6.2.3

Rewrite the expression.

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

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 $n = 4$ .

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

Step 2.1.3.3

Multiply $4$ by $- 2$ .

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

Step 2.2

Find the second derivative.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

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 $n = 4$ .

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

Step 2.2.2.3

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

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

Step 2.2.2.4

Multiply $4$ by $3$ .

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

Step 2.2.2.5

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

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

Step 2.2.2.6

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

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Step 2.2.2.6.1

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

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

Step 2.2.2.6.2

Cancel the common factors.

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Step 2.2.2.6.2.1

Factor $4$ out of $4$ .

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

Step 2.2.2.6.2.2

Cancel the common factor.

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

Step 2.2.2.6.2.3

Rewrite the expression.

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

Step 2.2.2.6.2.4

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

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

Step 2.2.3

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

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Step 2.2.3.1

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

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

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 $n = 3$ .

$3 x^{3} - 8 (3 x^{2})$

Step 2.2.3.3

Multiply $3$ by $- 8$ .

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

Step 2.3

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

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

Step 3

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

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Step 3.1

Set the second derivative equal to $0$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

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 $0$ .

$x^{2} = 0$

$x - 8 = 0$

Step 3.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 3.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 3.4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 3.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 3.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.5

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 3.5.2

Add $8$ to both sides of the equation.

$x = 8$

Step 3.6

The final solution is all the values that make $3 x^{2} (x - 8) = 0$ true.

$x = 0, 8$

Step 4

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

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Step 4.1

Substitute $0$ in $f (x) = \frac{3}{20} x^{5} - 2 x^{4}$ to find the value of $y$ .

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Step 4.1.1

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

$f (0) = \frac{3}{20} \cdot {(0)}^{5} - 2 {(0)}^{4}$

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 $0$ .

$f (0) = \frac{3}{20} \cdot 0 - 2 {(0)}^{4}$

Step 4.1.2.1.2

Multiply $\frac{3}{20}$ by $0$ .

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

Step 4.1.2.1.3

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

$f (0) = 0 - 2 \cdot 0$

Step 4.1.2.1.4

Multiply $- 2$ 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 $f (x) = \frac{3}{20} x^{5} - 2 x^{4}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4.3

Substitute $8$ in $f (x) = \frac{3}{20} x^{5} - 2 x^{4}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $8$ in the expression.

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

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 $8$ to the power of $5$ .

$f (8) = \frac{3}{20} \cdot 32768 - 2 {(8)}^{4}$

Step 4.3.2.1.2

Cancel the common factor of $4$ .

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Step 4.3.2.1.2.1

Factor $4$ out of $20$ .

$f (8) = \frac{3}{4 (5)} \cdot 32768 - 2 {(8)}^{4}$

Step 4.3.2.1.2.2

Factor $4$ out of $32768$ .

$f (8) = \frac{3}{4 \cdot 5} \cdot (4 \cdot 8192) - 2 {(8)}^{4}$

Step 4.3.2.1.2.3

Cancel the common factor.

$f (8) = \frac{3}{4 \cdot 5} \cdot (4 \cdot 8192) - 2 {(8)}^{4}$

Step 4.3.2.1.2.4

Rewrite the expression.

$f (8) = \frac{3}{5} \cdot 8192 - 2 {(8)}^{4}$

Step 4.3.2.1.3

Combine $\frac{3}{5}$ and $8192$ .

$f (8) = \frac{3 \cdot 8192}{5} - 2 {(8)}^{4}$

Step 4.3.2.1.4

Multiply $3$ by $8192$ .

$f (8) = \frac{24576}{5} - 2 {(8)}^{4}$

Step 4.3.2.1.5

Raise $8$ to the power of $4$ .

$f (8) = \frac{24576}{5} - 2 \cdot 4096$

Step 4.3.2.1.6

Multiply $- 2$ by $4096$ .

$f (8) = \frac{24576}{5} - 8192$

Step 4.3.2.2

To write $- 8192$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (8) = \frac{24576}{5} - 8192 \cdot \frac{5}{5}$

Step 4.3.2.3

Combine $- 8192$ and $\frac{5}{5}$ .

$f (8) = \frac{24576}{5} + \frac{- 8192 \cdot 5}{5}$

Step 4.3.2.4

Combine the numerators over the common denominator.

$f (8) = \frac{24576 - 8192 \cdot 5}{5}$

Step 4.3.2.5

Simplify the numerator.

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Step 4.3.2.5.1

Multiply $- 8192$ by $5$ .

$f (8) = \frac{24576 - 40960}{5}$

Step 4.3.2.5.2

Subtract $40960$ from $24576$ .

$f (8) = \frac{- 16384}{5}$

Step 4.3.2.6

Move the negative in front of the fraction.

$f (8) = - \frac{16384}{5}$

Step 4.3.2.7

The final answer is $- \frac{16384}{5}$ .

$- \frac{16384}{5}$

Step 4.4

The point found by substituting $8$ in $f (x) = \frac{3}{20} x^{5} - 2 x^{4}$ is $(8, - \frac{16384}{5})$ . This point can be an inflection point.

$(8, - \frac{16384}{5})$

Step 4.5

Determine the points that could be inflection points.

$(0, 0), (8, - \frac{16384}{5})$

Step 5

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

$(- \infty, 0) \cup (0, 8) \cup (8, \infty)$

Step 6

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

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Step 6.1

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

$f'' (- 0.1) = 3 {(- 0.1)}^{3} - 24 {(- 0.1)}^{2}$

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 $- 0.1$ to the power of $3$ .

$f'' (- 0.1) = 3 \cdot - 0.001 - 24 {(- 0.1)}^{2}$

Step 6.2.1.2

Multiply $3$ by $- 0.001$ .

$f'' (- 0.1) = - 0.003 - 24 {(- 0.1)}^{2}$

Step 6.2.1.3

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

$f'' (- 0.1) = - 0.003 - 24 \cdot 0.01$

Step 6.2.1.4

Multiply $- 24$ by $0.01$ .

$f'' (- 0.1) = - 0.003 - 0.24$

Step 6.2.2

Subtract $0.24$ from $- 0.003$ .

$f'' (- 0.1) = - 0.243$

Step 6.2.3

The final answer is $- 0.243$ .

$- 0.243$

Step 6.3

At $- 0.1$ , the second derivative is $- 0.243$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, 0)$

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

Step 7

Substitute a value from the interval $(0, 8)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

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

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

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 $4$ to the power of $3$ .

$f'' (4) = 3 \cdot 64 - 24 {(4)}^{2}$

Step 7.2.1.2

Multiply $3$ by $64$ .

$f'' (4) = 192 - 24 {(4)}^{2}$

Step 7.2.1.3

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

$f'' (4) = 192 - 24 \cdot 16$

Step 7.2.1.4

Multiply $- 24$ by $16$ .

$f'' (4) = 192 - 384$

Step 7.2.2

Subtract $384$ from $192$ .

$f'' (4) = - 192$

Step 7.2.3

The final answer is $- 192$ .

$- 192$

Step 7.3

At $4$ , the second derivative is $- 192$ . Since this is negative, the second derivative is decreasing on the interval $(0, 8)$

Decreasing on $(0, 8)$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(8, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $8.1$ in the expression.

$f'' (8.1) = 3 {(8.1)}^{3} - 24 {(8.1)}^{2}$

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 $8.1$ to the power of $3$ .

$f'' (8.1) = 3 \cdot 531.441 - 24 {(8.1)}^{2}$

Step 8.2.1.2

Multiply $3$ by $531.441$ .

$f'' (8.1) = 1594.323 - 24 {(8.1)}^{2}$

Step 8.2.1.3

Raise $8.1$ to the power of $2$ .

$f'' (8.1) = 1594.323 - 24 \cdot 65.61$

Step 8.2.1.4

Multiply $- 24$ by $65.61$ .

$f'' (8.1) = 1594.323 - 1574.64$

Step 8.2.2

Subtract $1574.64$ from $1594.323$ .

$f'' (8.1) = 19.683$

Step 8.2.3

The final answer is $19.683$ .

$19.683$

Step 8.3

At $8.1$ , the second derivative is $19.683$ . Since this is positive, the second derivative is increasing on the interval $(8, \infty)$ .

Increasing on $(8, \infty)$ 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 point in this case is $(8, - \frac{16384}{5})$ .

$(8, - \frac{16384}{5})$

Step 10