Calculus Examples

$- \frac{3}{20} x^{5} + 11 x^{3}$

Step 1

Write $- \frac{3}{20} x^{5} + 11 x^{3}$ as a function.

$f (x) = - \frac{3}{20} x^{5} + 11 x^{3}$

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 $- \frac{3}{20} x^{5} + 11 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- \frac{3}{20} x^{5}] + \frac{d}{d x} [11 x^{3}]$ .

$\frac{d}{d x} [- \frac{3}{20} x^{5}] + \frac{d}{d x} [11 x^{3}]$

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

$- \frac{3}{20} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [11 x^{3}]$

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 $n = 5$ .

$- \frac{3}{20} (5 x^{4}) + \frac{d}{d x} [11 x^{3}]$

Step 2.1.2.3

Multiply $5$ by $- 1$ .

$- 5 (\frac{3}{20}) x^{4} + \frac{d}{d x} [11 x^{3}]$

Step 2.1.2.4

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

$\frac{- 5 \cdot 3}{20} x^{4} + \frac{d}{d x} [11 x^{3}]$

Step 2.1.2.5

Multiply $- 5$ by $3$ .

$\frac{- 15}{20} x^{4} + \frac{d}{d x} [11 x^{3}]$

Step 2.1.2.6

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

$\frac{- 15 x^{4}}{20} + \frac{d}{d x} [11 x^{3}]$

Step 2.1.2.7

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

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Step 2.1.2.7.1

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

$\frac{5 (- 3 x^{4})}{20} + \frac{d}{d x} [11 x^{3}]$

Step 2.1.2.7.2

Cancel the common factors.

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Step 2.1.2.7.2.1

Factor $5$ out of $20$ .

$\frac{5 (- 3 x^{4})}{5 (4)} + \frac{d}{d x} [11 x^{3}]$

Step 2.1.2.7.2.2

Cancel the common factor.

$\frac{5 (- 3 x^{4})}{5 \cdot 4} + \frac{d}{d x} [11 x^{3}]$

Step 2.1.2.7.2.3

Rewrite the expression.

$\frac{- 3 x^{4}}{4} + \frac{d}{d x} [11 x^{3}]$

Step 2.1.2.8

Move the negative in front of the fraction.

$- \frac{3 x^{4}}{4} + \frac{d}{d x} [11 x^{3}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [11 x^{3}]$ .

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Step 2.1.3.1

Since $11$ is constant with respect to $x$ , the derivative of $11 x^{3}$ with respect to $x$ is $11 \frac{d}{d x} [x^{3}]$ .

$- \frac{3 x^{4}}{4} + 11 \frac{d}{d x} [x^{3}]$

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 $n = 3$ .

$- \frac{3 x^{4}}{4} + 11 (3 x^{2})$

Step 2.1.3.3

Multiply $3$ by $11$ .

$f' (x) = - \frac{3 x^{4}}{4} + 33 x^{2}$

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $- \frac{3 x^{4}}{4} + 33 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- \frac{3 x^{4}}{4}] + \frac{d}{d x} [33 x^{2}]$ .

$\frac{d}{d x} [- \frac{3 x^{4}}{4}] + \frac{d}{d x} [33 x^{2}]$

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

$- \frac{3}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [33 x^{2}]$

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 $n = 4$ .

$- \frac{3}{4} (4 x^{3}) + \frac{d}{d x} [33 x^{2}]$

Step 2.2.2.3

Multiply $4$ by $- 1$ .

$- 4 (\frac{3}{4}) x^{3} + \frac{d}{d x} [33 x^{2}]$

Step 2.2.2.4

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

$\frac{- 4 \cdot 3}{4} x^{3} + \frac{d}{d x} [33 x^{2}]$

Step 2.2.2.5

Multiply $- 4$ by $3$ .

$\frac{- 12}{4} x^{3} + \frac{d}{d x} [33 x^{2}]$

Step 2.2.2.6

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

$\frac{- 12 x^{3}}{4} + \frac{d}{d x} [33 x^{2}]$

Step 2.2.2.7

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

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Step 2.2.2.7.1

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

$\frac{4 (- 3 x^{3})}{4} + \frac{d}{d x} [33 x^{2}]$

Step 2.2.2.7.2

Cancel the common factors.

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Step 2.2.2.7.2.1

Factor $4$ out of $4$ .

$\frac{4 (- 3 x^{3})}{4 (1)} + \frac{d}{d x} [33 x^{2}]$

Step 2.2.2.7.2.2

Cancel the common factor.

$\frac{4 (- 3 x^{3})}{4 \cdot 1} + \frac{d}{d x} [33 x^{2}]$

Step 2.2.2.7.2.3

Rewrite the expression.

$\frac{- 3 x^{3}}{1} + \frac{d}{d x} [33 x^{2}]$

Step 2.2.2.7.2.4

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

$- 3 x^{3} + \frac{d}{d x} [33 x^{2}]$

Step 2.2.3

Evaluate $\frac{d}{d x} [33 x^{2}]$ .

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Step 2.2.3.1

Since $33$ is constant with respect to $x$ , the derivative of $33 x^{2}$ with respect to $x$ is $33 \frac{d}{d x} [x^{2}]$ .

$- 3 x^{3} + 33 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$- 3 x^{3} + 33 (2 x)$

Step 2.2.3.3

Multiply $2$ by $33$ .

$f'' (x) = - 3 x^{3} + 66 x$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $- 3 x^{3} + 66 x$ .

$- 3 x^{3} + 66 x$

Step 3

Set the second derivative equal to $0$ then solve the equation $- 3 x^{3} + 66 x = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$- 3 x^{3} + 66 x = 0$

Step 3.2

Factor $- 3 x$ out of $- 3 x^{3} + 66 x$ .

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Step 3.2.1

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

$- 3 x \cdot x^{2} + 66 x = 0$

Step 3.2.2

Factor $- 3 x$ out of $66 x$ .

$- 3 x \cdot x^{2} - 3 x \cdot - 22 = 0$

Step 3.2.3

Factor $- 3 x$ out of $- 3 x (x^{2}) - 3 x (- 22)$ .

$- 3 x (x^{2} - 22) = 0$

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 $0$ .

$x = 0$

$x^{2} - 22 = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $x^{2} - 22$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x^{2} - 22$ equal to $0$ .

$x^{2} - 22 = 0$

Step 3.5.2

Solve $x^{2} - 22 = 0$ for $x$ .

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Step 3.5.2.1

Add $22$ to both sides of the equation.

$x^{2} = 22$

Step 3.5.2.2

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

$x = \pm \sqrt{22}$

Step 3.5.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{22}$

Step 3.5.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{22}$

Step 3.5.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{22}, - \sqrt{22}$

Step 3.6

The final solution is all the values that make $- 3 x (x^{2} - 22) = 0$ true.

$x = 0, \sqrt{22}, - \sqrt{22}$

Step 4

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

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Step 4.1

Substitute $0$ in $f (x) = - \frac{3}{20} x^{5} + 11 x^{3}$ to find the value of $y$ .

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Step 4.1.1

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

$f (0) = - \frac{3}{20} \cdot {(0)}^{5} + 11 {(0)}^{3}$

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 $0$ .

$f (0) = - \frac{3}{20} \cdot 0 + 11 {(0)}^{3}$

Step 4.1.2.1.2

Multiply $- \frac{3}{20} \cdot 0$ .

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Step 4.1.2.1.2.1

Multiply $0$ by $- 1$ .

$f (0) = 0 (\frac{3}{20}) + 11 {(0)}^{3}$

Step 4.1.2.1.2.2

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

$f (0) = 0 + 11 {(0)}^{3}$

Step 4.1.2.1.3

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

$f (0) = 0 + 11 \cdot 0$

Step 4.1.2.1.4

Multiply $11$ 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 $f (x) = - \frac{3}{20} x^{5} + 11 x^{3}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4.3

Substitute $\sqrt{22}$ in $f (x) = - \frac{3}{20} x^{5} + 11 x^{3}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $\sqrt{22}$ in the expression.

$f (\sqrt{22}) = - \frac{3}{20} \cdot {(\sqrt{22})}^{5} + 11 {(\sqrt{22})}^{3}$

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

Rewrite ${\sqrt{22}}^{5}$ as $\sqrt{22^{5}}$ .

$f (\sqrt{22}) = - \frac{3}{20} \cdot \sqrt{22^{5}} + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.2

Raise $22$ to the power of $5$ .

$f (\sqrt{22}) = - \frac{3}{20} \cdot \sqrt{5153632} + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.3

Rewrite $5153632$ as $484^{2} \cdot 22$ .

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Step 4.3.2.1.3.1

Factor $234256$ out of $5153632$ .

$f (\sqrt{22}) = - \frac{3}{20} \cdot \sqrt{234256 (22)} + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.3.2

Rewrite $234256$ as $484^{2}$ .

$f (\sqrt{22}) = - \frac{3}{20} \cdot \sqrt{484^{2} \cdot 22} + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.4

Pull terms out from under the radical.

$f (\sqrt{22}) = - \frac{3}{20} \cdot (484 \sqrt{22}) + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.5

Cancel the common factor of $4$ .

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Step 4.3.2.1.5.1

Move the leading negative in $- \frac{3}{20}$ into the numerator.

$f (\sqrt{22}) = \frac{- 3}{20} \cdot (484 \sqrt{22}) + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.5.2

Factor $4$ out of $20$ .

$f (\sqrt{22}) = \frac{- 3}{4 (5)} \cdot (484 \sqrt{22}) + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.5.3

Factor $4$ out of $484 \sqrt{22}$ .

$f (\sqrt{22}) = \frac{- 3}{4 (5)} \cdot (4 (121 \sqrt{22})) + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.5.4

Cancel the common factor.

$f (\sqrt{22}) = \frac{- 3}{4 \cdot 5} \cdot (4 (121 \sqrt{22})) + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.5.5

Rewrite the expression.

$f (\sqrt{22}) = \frac{- 3}{5} \cdot (121 \sqrt{22}) + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.6

Combine $121$ and $\frac{- 3}{5}$ .

$f (\sqrt{22}) = \frac{121 \cdot - 3}{5} \cdot \sqrt{22} + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.7

Multiply $121$ by $- 3$ .

$f (\sqrt{22}) = \frac{- 363}{5} \cdot \sqrt{22} + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.8

Combine $\frac{- 363}{5}$ and $\sqrt{22}$ .

$f (\sqrt{22}) = \frac{- 363 \sqrt{22}}{5} + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.9

Move the negative in front of the fraction.

$f (\sqrt{22}) = - \frac{363 \sqrt{22}}{5} + 11 {(\sqrt{22})}^{3}$

Step 4.3.2.1.10

Rewrite ${\sqrt{22}}^{3}$ as $\sqrt{22^{3}}$ .

$f (\sqrt{22}) = - \frac{363 \sqrt{22}}{5} + 11 \sqrt{22^{3}}$

Step 4.3.2.1.11

Raise $22$ to the power of $3$ .

$f (\sqrt{22}) = - \frac{363 \sqrt{22}}{5} + 11 \sqrt{10648}$

Step 4.3.2.1.12

Rewrite $10648$ as $22^{2} \cdot 22$ .

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Step 4.3.2.1.12.1

Factor $484$ out of $10648$ .

$f (\sqrt{22}) = - \frac{363 \sqrt{22}}{5} + 11 \sqrt{484 (22)}$

Step 4.3.2.1.12.2

Rewrite $484$ as $22^{2}$ .

$f (\sqrt{22}) = - \frac{363 \sqrt{22}}{5} + 11 \sqrt{22^{2} \cdot 22}$

Step 4.3.2.1.13

Pull terms out from under the radical.

$f (\sqrt{22}) = - \frac{363 \sqrt{22}}{5} + 11 (22 \sqrt{22})$

Step 4.3.2.1.14

Multiply $22$ by $11$ .

$f (\sqrt{22}) = - \frac{363 \sqrt{22}}{5} + 242 \sqrt{22}$

Step 4.3.2.2

To write $242 \sqrt{22}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\sqrt{22}) = - \frac{363 \sqrt{22}}{5} + 242 \sqrt{22} \cdot \frac{5}{5}$

Step 4.3.2.3

Combine fractions.

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Step 4.3.2.3.1

Combine $242 \sqrt{22}$ and $\frac{5}{5}$ .

$f (\sqrt{22}) = - \frac{363 \sqrt{22}}{5} + \frac{242 \sqrt{22} \cdot 5}{5}$

Step 4.3.2.3.2

Combine the numerators over the common denominator.

$f (\sqrt{22}) = \frac{- 363 \sqrt{22} + 242 \sqrt{22} \cdot 5}{5}$

Step 4.3.2.4

Simplify the numerator.

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Step 4.3.2.4.1

Multiply $5$ by $242$ .

$f (\sqrt{22}) = \frac{- 363 \sqrt{22} + 1210 \sqrt{22}}{5}$

Step 4.3.2.4.2

Add $- 363 \sqrt{22}$ and $1210 \sqrt{22}$ .

$f (\sqrt{22}) = \frac{847 \sqrt{22}}{5}$

Step 4.3.2.5

The final answer is $\frac{847 \sqrt{22}}{5}$ .

$\frac{847 \sqrt{22}}{5}$

Step 4.4

The point found by substituting $\sqrt{22}$ in $f (x) = - \frac{3}{20} x^{5} + 11 x^{3}$ is $(\sqrt{22}, \frac{847 \sqrt{22}}{5})$ . This point can be an inflection point.

$(\sqrt{22}, \frac{847 \sqrt{22}}{5})$

Step 4.5

Substitute $- \sqrt{22}$ in $f (x) = - \frac{3}{20} x^{5} + 11 x^{3}$ to find the value of $y$ .

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Step 4.5.1

Replace the variable $x$ with $- \sqrt{22}$ in the expression.

$f (- \sqrt{22}) = - \frac{3}{20} \cdot {(- \sqrt{22})}^{5} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2

Simplify the result.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

Apply the product rule to $- \sqrt{22}$ .

$f (- \sqrt{22}) = - \frac{3}{20} \cdot ({(- 1)}^{5} {\sqrt{22}}^{5}) + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.2

Multiply $- 1$ by ${(- 1)}^{5}$ by adding the exponents.

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Step 4.5.2.1.2.1

Move ${(- 1)}^{5}$ .

$f (- \sqrt{22}) = {(- 1)}^{5} \cdot (- 1 (\frac{3}{20})) \cdot {\sqrt{22}}^{5} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.2.2

Multiply ${(- 1)}^{5}$ by $- 1$ .

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Step 4.5.2.1.2.2.1

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

$f (- \sqrt{22}) = {(- 1)}^{5} \cdot ((- 1) (\frac{3}{20})) \cdot {\sqrt{22}}^{5} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \sqrt{22}) = {(- 1)}^{5 + 1} (\frac{3}{20}) \cdot {\sqrt{22}}^{5} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.2.3

Add $5$ and $1$ .

$f (- \sqrt{22}) = {(- 1)}^{6} (\frac{3}{20}) \cdot {\sqrt{22}}^{5} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.3

Raise $- 1$ to the power of $6$ .

$f (- \sqrt{22}) = 1 (\frac{3}{20}) \cdot {\sqrt{22}}^{5} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.4

Multiply $\frac{3}{20}$ by $1$ .

$f (- \sqrt{22}) = \frac{3}{20} \cdot {\sqrt{22}}^{5} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.5

Rewrite ${\sqrt{22}}^{5}$ as $\sqrt{22^{5}}$ .

$f (- \sqrt{22}) = \frac{3}{20} \cdot \sqrt{22^{5}} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.6

Raise $22$ to the power of $5$ .

$f (- \sqrt{22}) = \frac{3}{20} \cdot \sqrt{5153632} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.7

Rewrite $5153632$ as $484^{2} \cdot 22$ .

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Step 4.5.2.1.7.1

Factor $234256$ out of $5153632$ .

$f (- \sqrt{22}) = \frac{3}{20} \cdot \sqrt{234256 (22)} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.7.2

Rewrite $234256$ as $484^{2}$ .

$f (- \sqrt{22}) = \frac{3}{20} \cdot \sqrt{484^{2} \cdot 22} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.8

Pull terms out from under the radical.

$f (- \sqrt{22}) = \frac{3}{20} \cdot (484 \sqrt{22}) + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.9

Cancel the common factor of $4$ .

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Step 4.5.2.1.9.1

Factor $4$ out of $20$ .

$f (- \sqrt{22}) = \frac{3}{4 (5)} \cdot (484 \sqrt{22}) + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.9.2

Factor $4$ out of $484 \sqrt{22}$ .

$f (- \sqrt{22}) = \frac{3}{4 (5)} \cdot (4 (121 \sqrt{22})) + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.9.3

Cancel the common factor.

$f (- \sqrt{22}) = \frac{3}{4 \cdot 5} \cdot (4 (121 \sqrt{22})) + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.9.4

Rewrite the expression.

$f (- \sqrt{22}) = \frac{3}{5} \cdot (121 \sqrt{22}) + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.10

Combine $121$ and $\frac{3}{5}$ .

$f (- \sqrt{22}) = \frac{121 \cdot 3}{5} \cdot \sqrt{22} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.11

Multiply $121$ by $3$ .

$f (- \sqrt{22}) = \frac{363}{5} \cdot \sqrt{22} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.12

Combine $\frac{363}{5}$ and $\sqrt{22}$ .

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} + 11 {(- \sqrt{22})}^{3}$

Step 4.5.2.1.13

Apply the product rule to $- \sqrt{22}$ .

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} + 11 ({(- 1)}^{3} {\sqrt{22}}^{3})$

Step 4.5.2.1.14

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

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} + 11 (- {\sqrt{22}}^{3})$

Step 4.5.2.1.15

Rewrite ${\sqrt{22}}^{3}$ as $\sqrt{22^{3}}$ .

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} + 11 (- \sqrt{22^{3}})$

Step 4.5.2.1.16

Raise $22$ to the power of $3$ .

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} + 11 (- \sqrt{10648})$

Step 4.5.2.1.17

Rewrite $10648$ as $22^{2} \cdot 22$ .

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Step 4.5.2.1.17.1

Factor $484$ out of $10648$ .

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} + 11 (- \sqrt{484 (22)})$

Step 4.5.2.1.17.2

Rewrite $484$ as $22^{2}$ .

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} + 11 (- \sqrt{22^{2} \cdot 22})$

Step 4.5.2.1.18

Pull terms out from under the radical.

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} + 11 (- (22 \sqrt{22}))$

Step 4.5.2.1.19

Multiply $22$ by $- 1$ .

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} + 11 (- 22 \sqrt{22})$

Step 4.5.2.1.20

Multiply $- 22$ by $11$ .

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} - 242 \sqrt{22}$

Step 4.5.2.2

To write $- 242 \sqrt{22}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} - 242 \sqrt{22} \cdot \frac{5}{5}$

Step 4.5.2.3

Combine fractions.

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Step 4.5.2.3.1

Combine $- 242 \sqrt{22}$ and $\frac{5}{5}$ .

$f (- \sqrt{22}) = \frac{363 \sqrt{22}}{5} + \frac{- 242 \sqrt{22} \cdot 5}{5}$

Step 4.5.2.3.2

Combine the numerators over the common denominator.

$f (- \sqrt{22}) = \frac{363 \sqrt{22} - 242 \sqrt{22} \cdot 5}{5}$

Step 4.5.2.4

Simplify the numerator.

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Step 4.5.2.4.1

Multiply $5$ by $- 242$ .

$f (- \sqrt{22}) = \frac{363 \sqrt{22} - 1210 \sqrt{22}}{5}$

Step 4.5.2.4.2

Subtract $1210 \sqrt{22}$ from $363 \sqrt{22}$ .

$f (- \sqrt{22}) = \frac{- 847 \sqrt{22}}{5}$

Step 4.5.2.5

Move the negative in front of the fraction.

$f (- \sqrt{22}) = - \frac{847 \sqrt{22}}{5}$

Step 4.5.2.6

The final answer is $- \frac{847 \sqrt{22}}{5}$ .

$- \frac{847 \sqrt{22}}{5}$

Step 4.6

The point found by substituting $- \sqrt{22}$ in $f (x) = - \frac{3}{20} x^{5} + 11 x^{3}$ is $(- \sqrt{22}, - \frac{847 \sqrt{22}}{5})$ . This point can be an inflection point.

$(- \sqrt{22}, - \frac{847 \sqrt{22}}{5})$

Step 4.7

Determine the points that could be inflection points.

$(0, 0), (\sqrt{22}, \frac{847 \sqrt{22}}{5}), (- \sqrt{22}, - \frac{847 \sqrt{22}}{5})$

Step 5

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

$(- \infty, - \sqrt{22}) \cup (- \sqrt{22}, 0) \cup (0, \sqrt{22}) \cup (\sqrt{22}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - \sqrt{22})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 4.79041575$ in the expression.

$f'' (- 4.79041575) = - 3 {(- 4.79041575)}^{3} + 66 (- 4.79041575)$

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 $- 4.79041575$ to the power of $3$ .

$f'' (- 4.79041575) = - 3 \cdot - 109.93085918 + 66 (- 4.79041575)$

Step 6.2.1.2

Multiply $- 3$ by $- 109.93085918$ .

$f'' (- 4.79041575) = 329.79257756 + 66 (- 4.79041575)$

Step 6.2.1.3

Multiply $66$ by $- 4.79041575$ .

$f'' (- 4.79041575) = 329.79257756 - 316.16744014$

Step 6.2.2

Subtract $316.16744014$ from $329.79257756$ .

$f'' (- 4.79041575) = 13.62513741$

Step 6.2.3

The final answer is $13.62513741$ .

$13.62513741$

Step 6.3

At $- 4.79041575$ , the second derivative is $13.62513741$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - \sqrt{22})$ .

Increasing on $(- \infty, - \sqrt{22})$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(- \sqrt{22}, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 2.34520787$ in the expression.

$f'' (- 2.34520787) = - 3 {(- 2.34520787)}^{3} + 66 (- 2.34520787)$

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 $- 2.34520787$ to the power of $3$ .

$f'' (- 2.34520787) = - 3 \cdot - 12.89864333 + 66 (- 2.34520787)$

Step 7.2.1.2

Multiply $- 3$ by $- 12.89864333$ .

$f'' (- 2.34520787) = 38.69593001 + 66 (- 2.34520787)$

Step 7.2.1.3

Multiply $66$ by $- 2.34520787$ .

$f'' (- 2.34520787) = 38.69593001 - 154.78372007$

Step 7.2.2

Subtract $154.78372007$ from $38.69593001$ .

$f'' (- 2.34520787) = - 116.08779005$

Step 7.2.3

The final answer is $- 116.08779005$ .

$- 116.08779005$

Step 7.3

At $- 2.34520787$ , the second derivative is $- 116.08779005$ . Since this is negative, the second derivative is decreasing on the interval $(- \sqrt{22}, 0)$

Decreasing on $(- \sqrt{22}, 0)$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(0, \sqrt{22})$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $2.34520787$ in the expression.

$f'' (2.34520787) = - 3 {(2.34520787)}^{3} + 66 (2.34520787)$

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 $2.34520787$ to the power of $3$ .

$f'' (2.34520787) = - 3 \cdot 12.89864333 + 66 (2.34520787)$

Step 8.2.1.2

Multiply $- 3$ by $12.89864333$ .

$f'' (2.34520787) = - 38.69593001 + 66 (2.34520787)$

Step 8.2.1.3

Multiply $66$ by $2.34520787$ .

$f'' (2.34520787) = - 38.69593001 + 154.78372007$

Step 8.2.2

Add $- 38.69593001$ and $154.78372007$ .

$f'' (2.34520787) = 116.08779005$

Step 8.2.3

The final answer is $116.08779005$ .

$116.08779005$

Step 8.3

At $2.34520787$ , the second derivative is $116.08779005$ . Since this is positive, the second derivative is increasing on the interval $(0, \sqrt{22})$ .

Increasing on $(0, \sqrt{22})$ since $f'' (x) > 0$

Step 9

Substitute a value from the interval $(\sqrt{22}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $4.79041575$ in the expression.

$f'' (4.79041575) = - 3 {(4.79041575)}^{3} + 66 (4.79041575)$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Raise $4.79041575$ to the power of $3$ .

$f'' (4.79041575) = - 3 \cdot 109.93085918 + 66 (4.79041575)$

Step 9.2.1.2

Multiply $- 3$ by $109.93085918$ .

$f'' (4.79041575) = - 329.79257756 + 66 (4.79041575)$

Step 9.2.1.3

Multiply $66$ by $4.79041575$ .

$f'' (4.79041575) = - 329.79257756 + 316.16744014$

Step 9.2.2

Add $- 329.79257756$ and $316.16744014$ .

$f'' (4.79041575) = - 13.62513741$

Step 9.2.3

The final answer is $- 13.62513741$ .

$- 13.62513741$

Step 9.3

At $4.79041575$ , the second derivative is $- 13.62513741$ . Since this is negative, the second derivative is decreasing on the interval $(\sqrt{22}, \infty)$

Decreasing on $(\sqrt{22}, \infty)$ since $f'' (x) < 0$

Step 10

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 $(- \sqrt{22}, - \frac{847 \sqrt{22}}{5}), (0, 0), (\sqrt{22}, \frac{847 \sqrt{22}}{5})$ .

$(- \sqrt{22}, - \frac{847 \sqrt{22}}{5}), (0, 0), (\sqrt{22}, \frac{847 \sqrt{22}}{5})$

Step 11