Calculus Examples

$f (x) = - \frac{1}{6} x^{6} - 2 x^{5}$

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 $- \frac{1}{6} x^{6} - 2 x^{5}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{6} x^{6}] + \frac{d}{d x} [- 2 x^{5}]$ .

$\frac{d}{d x} [- \frac{1}{6} x^{6}] + \frac{d}{d x} [- 2 x^{5}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- \frac{1}{6} x^{6}]$ .

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Step 1.1.2.1

Since $- \frac{1}{6}$ is constant with respect to $x$ , the derivative of $- \frac{1}{6} x^{6}$ with respect to $x$ is $- \frac{1}{6} \frac{d}{d x} [x^{6}]$ .

$- \frac{1}{6} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 2 x^{5}]$

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 $n = 6$ .

$- \frac{1}{6} (6 x^{5}) + \frac{d}{d x} [- 2 x^{5}]$

Step 1.1.2.3

Multiply $6$ by $- 1$ .

$- 6 (\frac{1}{6}) x^{5} + \frac{d}{d x} [- 2 x^{5}]$

Step 1.1.2.4

Combine $- 6$ and $\frac{1}{6}$ .

$\frac{- 6}{6} x^{5} + \frac{d}{d x} [- 2 x^{5}]$

Step 1.1.2.5

Combine $\frac{- 6}{6}$ and $x^{5}$ .

$\frac{- 6 x^{5}}{6} + \frac{d}{d x} [- 2 x^{5}]$

Step 1.1.2.6

Cancel the common factor of $- 6$ and $6$ .

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Step 1.1.2.6.1

Factor $6$ out of $- 6 x^{5}$ .

$\frac{6 (- x^{5})}{6} + \frac{d}{d x} [- 2 x^{5}]$

Step 1.1.2.6.2

Cancel the common factors.

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Step 1.1.2.6.2.1

Factor $6$ out of $6$ .

$\frac{6 (- x^{5})}{6 (1)} + \frac{d}{d x} [- 2 x^{5}]$

Step 1.1.2.6.2.2

Cancel the common factor.

$\frac{6 (- x^{5})}{6 \cdot 1} + \frac{d}{d x} [- 2 x^{5}]$

Step 1.1.2.6.2.3

Rewrite the expression.

$\frac{- x^{5}}{1} + \frac{d}{d x} [- 2 x^{5}]$

Step 1.1.2.6.2.4

Divide $- x^{5}$ by $1$ .

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

Step 1.1.3

Evaluate $\frac{d}{d x} [- 2 x^{5}]$ .

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Step 1.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{5}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{5}]$ .

$- x^{5} - 2 \frac{d}{d x} [x^{5}]$

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 $n = 5$ .

$- x^{5} - 2 (5 x^{4})$

Step 1.1.3.3

Multiply $5$ by $- 2$ .

$f' (x) = - x^{5} - 10 x^{4}$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $- x^{5} - 10 x^{4}$ with respect to $x$ is $\frac{d}{d x} [- x^{5}] + \frac{d}{d x} [- 10 x^{4}]$ .

$\frac{d}{d x} [- x^{5}] + \frac{d}{d x} [- 10 x^{4}]$

Step 1.2.2

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

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Step 1.2.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{5}$ with respect to $x$ is $- \frac{d}{d x} [x^{5}]$ .

$- \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 10 x^{4}]$

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 $n = 5$ .

$- (5 x^{4}) + \frac{d}{d x} [- 10 x^{4}]$

Step 1.2.2.3

Multiply $5$ by $- 1$ .

$- 5 x^{4} + \frac{d}{d x} [- 10 x^{4}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 10 x^{4}]$ .

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Step 1.2.3.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x^{4}$ with respect to $x$ is $- 10 \frac{d}{d x} [x^{4}]$ .

$- 5 x^{4} - 10 \frac{d}{d x} [x^{4}]$

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 $n = 4$ .

$- 5 x^{4} - 10 (4 x^{3})$

Step 1.2.3.3

Multiply $4$ by $- 10$ .

$f'' (x) = - 5 x^{4} - 40 x^{3}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- 5 x^{4} - 40 x^{3}$ .

$- 5 x^{4} - 40 x^{3}$

Step 2

Set the second derivative equal to $0$ then solve the equation $- 5 x^{4} - 40 x^{3} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- 5 x^{4} - 40 x^{3} = 0$

Step 2.2

Factor $- 5 x^{3}$ out of $- 5 x^{4} - 40 x^{3}$ .

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Step 2.2.1

Factor $- 5 x^{3}$ out of $- 5 x^{4}$ .

$- 5 x^{3} x - 40 x^{3} = 0$

Step 2.2.2

Factor $- 5 x^{3}$ out of $- 40 x^{3}$ .

$- 5 x^{3} x - 5 x^{3} \cdot 8 = 0$

Step 2.2.3

Factor $- 5 x^{3}$ out of $- 5 x^{3} (x) - 5 x^{3} (8)$ .

$- 5 x^{3} (x + 8) = 0$

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 $0$ .

$x^{3} = 0$

$x + 8 = 0$

Step 2.4

Set $x^{3}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Step 2.4.2

Solve $x^{3} = 0$ for $x$ .

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Step 2.4.2.1

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

$x = \sqrt[3]{0}$

Step 2.4.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 2.4.2.2.2

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

$x = 0$

Step 2.5

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

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Step 2.5.1

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

$x + 8 = 0$

Step 2.5.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 2.6

The final solution is all the values that make $- 5 x^{3} (x + 8) = 0$ true.

$x = 0, - 8$

Step 3

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

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Step 3.1

Substitute $0$ in $f (x) = - \frac{1}{6} x^{6} - 2 x^{5}$ to find the value of $y$ .

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Step 3.1.1

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

$f (0) = - \frac{1}{6} \cdot {(0)}^{6} - 2 {(0)}^{5}$

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 $0$ .

$f (0) = - \frac{1}{6} \cdot 0 - 2 {(0)}^{5}$

Step 3.1.2.1.2

Multiply $- \frac{1}{6} \cdot 0$ .

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Step 3.1.2.1.2.1

Multiply $0$ by $- 1$ .

$f (0) = 0 (\frac{1}{6}) - 2 {(0)}^{5}$

Step 3.1.2.1.2.2

Multiply $0$ by $\frac{1}{6}$ .

$f (0) = 0 - 2 {(0)}^{5}$

Step 3.1.2.1.3

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

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

Step 3.1.2.1.4

Multiply $- 2$ 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 $f (x) = - \frac{1}{6} x^{6} - 2 x^{5}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.3

Substitute $- 8$ in $f (x) = - \frac{1}{6} x^{6} - 2 x^{5}$ to find the value of $y$ .

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Step 3.3.1

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

$f (- 8) = - \frac{1}{6} \cdot {(- 8)}^{6} - 2 {(- 8)}^{5}$

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 $- 8$ to the power of $6$ .

$f (- 8) = - \frac{1}{6} \cdot 262144 - 2 {(- 8)}^{5}$

Step 3.3.2.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.2.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$f (- 8) = \frac{- 1}{6} \cdot 262144 - 2 {(- 8)}^{5}$

Step 3.3.2.1.2.2

Factor $2$ out of $6$ .

$f (- 8) = \frac{- 1}{2 (3)} \cdot 262144 - 2 {(- 8)}^{5}$

Step 3.3.2.1.2.3

Factor $2$ out of $262144$ .

$f (- 8) = \frac{- 1}{2 \cdot 3} \cdot (2 \cdot 131072) - 2 {(- 8)}^{5}$

Step 3.3.2.1.2.4

Cancel the common factor.

$f (- 8) = \frac{- 1}{2 \cdot 3} \cdot (2 \cdot 131072) - 2 {(- 8)}^{5}$

Step 3.3.2.1.2.5

Rewrite the expression.

$f (- 8) = \frac{- 1}{3} \cdot 131072 - 2 {(- 8)}^{5}$

Step 3.3.2.1.3

Combine $\frac{- 1}{3}$ and $131072$ .

$f (- 8) = \frac{- 1 \cdot 131072}{3} - 2 {(- 8)}^{5}$

Step 3.3.2.1.4

Multiply $- 1$ by $131072$ .

$f (- 8) = \frac{- 131072}{3} - 2 {(- 8)}^{5}$

Step 3.3.2.1.5

Move the negative in front of the fraction.

$f (- 8) = - \frac{131072}{3} - 2 {(- 8)}^{5}$

Step 3.3.2.1.6

Raise $- 8$ to the power of $5$ .

$f (- 8) = - \frac{131072}{3} - 2 \cdot - 32768$

Step 3.3.2.1.7

Multiply $- 2$ by $- 32768$ .

$f (- 8) = - \frac{131072}{3} + 65536$

Step 3.3.2.2

To write $65536$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- 8) = - \frac{131072}{3} + 65536 \cdot \frac{3}{3}$

Step 3.3.2.3

Combine $65536$ and $\frac{3}{3}$ .

$f (- 8) = - \frac{131072}{3} + \frac{65536 \cdot 3}{3}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$f (- 8) = \frac{- 131072 + 65536 \cdot 3}{3}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $65536$ by $3$ .

$f (- 8) = \frac{- 131072 + 196608}{3}$

Step 3.3.2.5.2

Add $- 131072$ and $196608$ .

$f (- 8) = \frac{65536}{3}$

Step 3.3.2.6

The final answer is $\frac{65536}{3}$ .

$\frac{65536}{3}$

Step 3.4

The point found by substituting $- 8$ in $f (x) = - \frac{1}{6} x^{6} - 2 x^{5}$ is $(- 8, \frac{65536}{3})$ . This point can be an inflection point.

$(- 8, \frac{65536}{3})$

Step 3.5

Determine the points that could be inflection points.

$(0, 0), (- 8, \frac{65536}{3})$

Step 4

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

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

Step 5

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

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Step 5.1

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

$f'' (- 8.1) = - 5 {(- 8.1)}^{4} - 40 {(- 8.1)}^{3}$

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 $- 8.1$ to the power of $4$ .

$f'' (- 8.1) = - 5 \cdot 4304.6721 - 40 {(- 8.1)}^{3}$

Step 5.2.1.2

Multiply $- 5$ by $4304.6721$ .

$f'' (- 8.1) = - 21523.3605 - 40 {(- 8.1)}^{3}$

Step 5.2.1.3

Raise $- 8.1$ to the power of $3$ .

$f'' (- 8.1) = - 21523.3605 - 40 \cdot - 531.441$

Step 5.2.1.4

Multiply $- 40$ by $- 531.441$ .

$f'' (- 8.1) = - 21523.3605 + 21257.64$

Step 5.2.2

Add $- 21523.3605$ and $21257.64$ .

$f'' (- 8.1) = - 265.7205$

Step 5.2.3

The final answer is $- 265.7205$ .

$- 265.7205$

Step 5.3

At $- 8.1$ , the second derivative is $- 265.7205$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 8)$

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

Step 6

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

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Step 6.1

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

$f'' (- 4) = - 5 {(- 4)}^{4} - 40 {(- 4)}^{3}$

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

$f'' (- 4) = - 5 \cdot 256 - 40 {(- 4)}^{3}$

Step 6.2.1.2

Multiply $- 5$ by $256$ .

$f'' (- 4) = - 1280 - 40 {(- 4)}^{3}$

Step 6.2.1.3

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

$f'' (- 4) = - 1280 - 40 \cdot - 64$

Step 6.2.1.4

Multiply $- 40$ by $- 64$ .

$f'' (- 4) = - 1280 + 2560$

Step 6.2.2

Add $- 1280$ and $2560$ .

$f'' (- 4) = 1280$

Step 6.2.3

The final answer is $1280$ .

$1280$

Step 6.3

At $- 4$ , the second derivative is $1280$ . Since this is positive, the second derivative is increasing on the interval $(- 8, 0)$ .

Increasing on $(- 8, 0)$ 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.

$f'' (0.1) = - 5 {(0.1)}^{4} - 40 {(0.1)}^{3}$

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 $4$ .

$f'' (0.1) = - 5 \cdot 0.0001 - 40 {(0.1)}^{3}$

Step 7.2.1.2

Multiply $- 5$ by $0.0001$ .

$f'' (0.1) = - 0.0005 - 40 {(0.1)}^{3}$

Step 7.2.1.3

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

$f'' (0.1) = - 0.0005 - 40 \cdot 0.001$

Step 7.2.1.4

Multiply $- 40$ by $0.001$ .

$f'' (0.1) = - 0.0005 - 0.04$

Step 7.2.2

Subtract $0.04$ from $- 0.0005$ .

$f'' (0.1) = - 0.0405$

Step 7.2.3

The final answer is $- 0.0405$ .

$- 0.0405$

Step 7.3

At $0.1$ , the second derivative is $- 0.0405$ . Since this is negative, the second derivative is decreasing on the interval $(0, \infty)$

Decreasing 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 $(- 8, \frac{65536}{3}), (0, 0)$ .

$(- 8, \frac{65536}{3}), (0, 0)$

Step 9