Calculus Examples

$f (x) = 6 x^{4} + 32 x^{3}$

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

By the Sum Rule, the derivative of $6 x^{4} + 32 x^{3}$ with respect to $x$ is $\frac{d}{d x} [6 x^{4}] + \frac{d}{d x} [32 x^{3}]$ .

$\frac{d}{d x} [6 x^{4}] + \frac{d}{d x} [32 x^{3}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [6 x^{4}]$ .

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Step 1.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{4}$ with respect to $x$ is $6 \frac{d}{d x} [x^{4}]$ .

$6 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [32 x^{3}]$

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 $n = 4$ .

$6 (4 x^{3}) + \frac{d}{d x} [32 x^{3}]$

Step 1.1.2.3

Multiply $4$ by $6$ .

$24 x^{3} + \frac{d}{d x} [32 x^{3}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [32 x^{3}]$ .

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Step 1.1.3.1

Since $32$ is constant with respect to $x$ , the derivative of $32 x^{3}$ with respect to $x$ is $32 \frac{d}{d x} [x^{3}]$ .

$24 x^{3} + 32 \frac{d}{d x} [x^{3}]$

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 $n = 3$ .

$24 x^{3} + 32 (3 x^{2})$

Step 1.1.3.3

Multiply $3$ by $32$ .

$f' (x) = 24 x^{3} + 96 x^{2}$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $24 x^{3} + 96 x^{2}$ with respect to $x$ is $\frac{d}{d x} [24 x^{3}] + \frac{d}{d x} [96 x^{2}]$ .

$\frac{d}{d x} [24 x^{3}] + \frac{d}{d x} [96 x^{2}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [24 x^{3}]$ .

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Step 1.2.2.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x^{3}$ with respect to $x$ is $24 \frac{d}{d x} [x^{3}]$ .

$24 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [96 x^{2}]$

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 $n = 3$ .

$24 (3 x^{2}) + \frac{d}{d x} [96 x^{2}]$

Step 1.2.2.3

Multiply $3$ by $24$ .

$72 x^{2} + \frac{d}{d x} [96 x^{2}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [96 x^{2}]$ .

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Step 1.2.3.1

Since $96$ is constant with respect to $x$ , the derivative of $96 x^{2}$ with respect to $x$ is $96 \frac{d}{d x} [x^{2}]$ .

$72 x^{2} + 96 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$72 x^{2} + 96 (2 x)$

Step 1.2.3.3

Multiply $2$ by $96$ .

$f'' (x) = 72 x^{2} + 192 x$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $72 x^{2} + 192 x$ .

$72 x^{2} + 192 x$

Step 2

Set the second derivative equal to $0$ then solve the equation $72 x^{2} + 192 x = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$72 x^{2} + 192 x = 0$

Step 2.2

Factor $24 x$ out of $72 x^{2} + 192 x$ .

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Step 2.2.1

Factor $24 x$ out of $72 x^{2}$ .

$24 x (3 x) + 192 x = 0$

Step 2.2.2

Factor $24 x$ out of $192 x$ .

$24 x (3 x) + 24 x (8) = 0$

Step 2.2.3

Factor $24 x$ out of $24 x (3 x) + 24 x (8)$ .

$24 x (3 x + 8) = 0$

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 $0$ .

$x = 0$

$3 x + 8 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $3 x + 8$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $3 x + 8$ equal to $0$ .

$3 x + 8 = 0$

Step 2.5.2

Solve $3 x + 8 = 0$ for $x$ .

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Step 2.5.2.1

Subtract $8$ from both sides of the equation.

$3 x = - 8$

Step 2.5.2.2

Divide each term in $3 x = - 8$ by $3$ and simplify.

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Step 2.5.2.2.1

Divide each term in $3 x = - 8$ by $3$ .

$\frac{3 x}{3} = \frac{- 8}{3}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 8}{3}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{3}$

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{8}{3}$

Step 2.6

The final solution is all the values that make $24 x (3 x + 8) = 0$ true.

$x = 0, - \frac{8}{3}$

Step 3

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

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Step 3.1

Substitute $0$ in $f (x) = 6 x^{4} + 32 x^{3}$ to find the value of $y$ .

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Step 3.1.1

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

$f (0) = 6 {(0)}^{4} + 32 {(0)}^{3}$

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 $0$ .

$f (0) = 6 \cdot 0 + 32 {(0)}^{3}$

Step 3.1.2.1.2

Multiply $6$ by $0$ .

$f (0) = 0 + 32 {(0)}^{3}$

Step 3.1.2.1.3

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

$f (0) = 0 + 32 \cdot 0$

Step 3.1.2.1.4

Multiply $32$ 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 $f (x) = 6 x^{4} + 32 x^{3}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.3

Substitute $- \frac{8}{3}$ in $f (x) = 6 x^{4} + 32 x^{3}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- \frac{8}{3}$ in the expression.

$f (- \frac{8}{3}) = 6 {(- \frac{8}{3})}^{4} + 32 {(- \frac{8}{3})}^{3}$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.2.1.1.1

Apply the product rule to $- \frac{8}{3}$ .

$f (- \frac{8}{3}) = 6 ({(- 1)}^{4} {(\frac{8}{3})}^{4}) + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.1.2

Apply the product rule to $\frac{8}{3}$ .

$f (- \frac{8}{3}) = 6 ({(- 1)}^{4} (\frac{8^{4}}{3^{4}})) + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.2

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

$f (- \frac{8}{3}) = 6 (1 (\frac{8^{4}}{3^{4}})) + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.3

Multiply $\frac{8^{4}}{3^{4}}$ by $1$ .

$f (- \frac{8}{3}) = 6 (\frac{8^{4}}{3^{4}}) + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.4

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

$f (- \frac{8}{3}) = 6 (\frac{4096}{3^{4}}) + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.5

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

$f (- \frac{8}{3}) = 6 (\frac{4096}{81}) + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.6

Cancel the common factor of $3$ .

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Step 3.3.2.1.6.1

Factor $3$ out of $6$ .

$f (- \frac{8}{3}) = 3 (2) (\frac{4096}{81}) + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.6.2

Factor $3$ out of $81$ .

$f (- \frac{8}{3}) = 3 \cdot (2 (\frac{4096}{3 \cdot 27})) + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.6.3

Cancel the common factor.

$f (- \frac{8}{3}) = 3 \cdot (2 (\frac{4096}{3 \cdot 27})) + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.6.4

Rewrite the expression.

$f (- \frac{8}{3}) = 2 (\frac{4096}{27}) + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.7

Combine $2$ and $\frac{4096}{27}$ .

$f (- \frac{8}{3}) = \frac{2 \cdot 4096}{27} + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.8

Multiply $2$ by $4096$ .

$f (- \frac{8}{3}) = \frac{8192}{27} + 32 {(- \frac{8}{3})}^{3}$

Step 3.3.2.1.9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.2.1.9.1

Apply the product rule to $- \frac{8}{3}$ .

$f (- \frac{8}{3}) = \frac{8192}{27} + 32 ({(- 1)}^{3} {(\frac{8}{3})}^{3})$

Step 3.3.2.1.9.2

Apply the product rule to $\frac{8}{3}$ .

$f (- \frac{8}{3}) = \frac{8192}{27} + 32 ({(- 1)}^{3} (\frac{8^{3}}{3^{3}}))$

Step 3.3.2.1.10

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

$f (- \frac{8}{3}) = \frac{8192}{27} + 32 (- \frac{8^{3}}{3^{3}})$

Step 3.3.2.1.11

Raise $8$ to the power of $3$ .

$f (- \frac{8}{3}) = \frac{8192}{27} + 32 (- \frac{512}{3^{3}})$

Step 3.3.2.1.12

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

$f (- \frac{8}{3}) = \frac{8192}{27} + 32 (- \frac{512}{27})$

Step 3.3.2.1.13

Multiply $32 (- \frac{512}{27})$ .

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Step 3.3.2.1.13.1

Multiply $- 1$ by $32$ .

$f (- \frac{8}{3}) = \frac{8192}{27} - 32 (\frac{512}{27})$

Step 3.3.2.1.13.2

Combine $- 32$ and $\frac{512}{27}$ .

$f (- \frac{8}{3}) = \frac{8192}{27} + \frac{- 32 \cdot 512}{27}$

Step 3.3.2.1.13.3

Multiply $- 32$ by $512$ .

$f (- \frac{8}{3}) = \frac{8192}{27} + \frac{- 16384}{27}$

Step 3.3.2.1.14

Move the negative in front of the fraction.

$f (- \frac{8}{3}) = \frac{8192}{27} - \frac{16384}{27}$

Step 3.3.2.2

Combine fractions.

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Step 3.3.2.2.1

Combine the numerators over the common denominator.

$f (- \frac{8}{3}) = \frac{8192 - 16384}{27}$

Step 3.3.2.2.2

Simplify the expression.

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Step 3.3.2.2.2.1

Subtract $16384$ from $8192$ .

$f (- \frac{8}{3}) = \frac{- 8192}{27}$

Step 3.3.2.2.2.2

Move the negative in front of the fraction.

$f (- \frac{8}{3}) = - \frac{8192}{27}$

Step 3.3.2.3

The final answer is $- \frac{8192}{27}$ .

$- \frac{8192}{27}$

Step 3.4

The point found by substituting $- \frac{8}{3}$ in $f (x) = 6 x^{4} + 32 x^{3}$ is $(- \frac{8}{3}, - \frac{8192}{27})$ . This point can be an inflection point.

$(- \frac{8}{3}, - \frac{8192}{27})$

Step 3.5

Determine the points that could be inflection points.

$(0, 0), (- \frac{8}{3}, - \frac{8192}{27})$

Step 4

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

$(- \infty, - \frac{8}{3}) \cup (- \frac{8}{3}, 0) \cup (0, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \frac{8}{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 2.7\overline{6}$ in the expression.

$f'' (- 2.7\overline{6}) = 72 {(- 2.7\overline{6})}^{2} + 192 (- 2.7\overline{6})$

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 $- 2.7\overline{6}$ to the power of $2$ .

$f'' (- 2.7\overline{6}) = 72 \cdot 7.65\overline{4} + 192 (- 2.7\overline{6})$

Step 5.2.1.2

Multiply $72$ by $7.65\overline{4}$ .

$f'' (- 2.7\overline{6}) = 551.12 + 192 (- 2.7\overline{6})$

Step 5.2.1.3

Multiply $192$ by $- 2.7\overline{6}$ .

$f'' (- 2.7\overline{6}) = 551.12 - 531.2$

Step 5.2.2

Subtract $531.2$ from $551.12$ .

$f'' (- 2.7\overline{6}) = 19.92$

Step 5.2.3

The final answer is $19.92$ .

$19.92$

Step 5.3

At $- 2.7\overline{6}$ , the second derivative is $19.92$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - \frac{8}{3})$ .

Increasing on $(- \infty, - \frac{8}{3})$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- \frac{8}{3}, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.33333333$ in the expression.

$f'' (- 1.33333333) = 72 {(- 1.33333333)}^{2} + 192 (- 1.33333333)$

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 $- 1.33333333$ to the power of $2$ .

$f'' (- 1.33333333) = 72 \cdot 1.\overline{7} + 192 (- 1.33333333)$

Step 6.2.1.2

Multiply $72$ by $1.\overline{7}$ .

$f'' (- 1.33333333) = 128 + 192 (- 1.33333333)$

Step 6.2.1.3

Multiply $192$ by $- 1.33333333$ .

$f'' (- 1.33333333) = 128 - 256$

Step 6.2.2

Subtract $256$ from $128$ .

$f'' (- 1.33333333) = - 128$

Step 6.2.3

The final answer is $- 128$ .

$- 128$

Step 6.3

At $- 1.33333333$ , the second derivative is $- 128$ . Since this is negative, the second derivative is decreasing on the interval $(- \frac{8}{3}, 0)$

Decreasing on $(- \frac{8}{3}, 0)$ 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.

$f'' (0.1) = 72 {(0.1)}^{2} + 192 (0.1)$

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 $2$ .

$f'' (0.1) = 72 \cdot 0.01 + 192 (0.1)$

Step 7.2.1.2

Multiply $72$ by $0.01$ .

$f'' (0.1) = 0.72 + 192 (0.1)$

Step 7.2.1.3

Multiply $192$ by $0.1$ .

$f'' (0.1) = 0.72 + 19.2$

Step 7.2.2

Add $0.72$ and $19.2$ .

$f'' (0.1) = 19.92$

Step 7.2.3

The final answer is $19.92$ .

$19.92$

Step 7.3

At $0.1$ , the second derivative is $19.92$ . 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 $(- \frac{8}{3}, - \frac{8192}{27}), (0, 0)$ .

$(- \frac{8}{3}, - \frac{8192}{27}), (0, 0)$

Step 9