Calculus Examples

$f (x) = - \frac{1}{2} x^{4} + 42 x^{2}$

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 $- \frac{1}{2} x^{4} + 42 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{2} x^{4}] + \frac{d}{d x} [42 x^{2}]$ .

$\frac{d}{d x} [- \frac{1}{2} x^{4}] + \frac{d}{d x} [42 x^{2}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- \frac{1}{2} x^{4}]$ .

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Step 1.1.2.1

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{1}{2} x^{4}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x^{4}]$ .

$- \frac{1}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [42 x^{2}]$

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 $n = 4$ .

$- \frac{1}{2} (4 x^{3}) + \frac{d}{d x} [42 x^{2}]$

Step 1.1.2.3

Multiply $4$ by $- 1$ .

$- 4 (\frac{1}{2}) x^{3} + \frac{d}{d x} [42 x^{2}]$

Step 1.1.2.4

Combine $- 4$ and $\frac{1}{2}$ .

$\frac{- 4}{2} x^{3} + \frac{d}{d x} [42 x^{2}]$

Step 1.1.2.5

Combine $\frac{- 4}{2}$ and $x^{3}$ .

$\frac{- 4 x^{3}}{2} + \frac{d}{d x} [42 x^{2}]$

Step 1.1.2.6

Cancel the common factor of $- 4$ and $2$ .

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Step 1.1.2.6.1

Factor $2$ out of $- 4 x^{3}$ .

$\frac{2 (- 2 x^{3})}{2} + \frac{d}{d x} [42 x^{2}]$

Step 1.1.2.6.2

Cancel the common factors.

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Step 1.1.2.6.2.1

Factor $2$ out of $2$ .

$\frac{2 (- 2 x^{3})}{2 (1)} + \frac{d}{d x} [42 x^{2}]$

Step 1.1.2.6.2.2

Cancel the common factor.

$\frac{2 (- 2 x^{3})}{2 \cdot 1} + \frac{d}{d x} [42 x^{2}]$

Step 1.1.2.6.2.3

Rewrite the expression.

$\frac{- 2 x^{3}}{1} + \frac{d}{d x} [42 x^{2}]$

Step 1.1.2.6.2.4

Divide $- 2 x^{3}$ by $1$ .

$- 2 x^{3} + \frac{d}{d x} [42 x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [42 x^{2}]$ .

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Step 1.1.3.1

Since $42$ is constant with respect to $x$ , the derivative of $42 x^{2}$ with respect to $x$ is $42 \frac{d}{d x} [x^{2}]$ .

$- 2 x^{3} + 42 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$- 2 x^{3} + 42 (2 x)$

Step 1.1.3.3

Multiply $2$ by $42$ .

$f' (x) = - 2 x^{3} + 84 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $- 2 x^{3} + 84 x$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [84 x]$ .

$\frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [84 x]$

Step 1.2.2

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

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Step 1.2.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

$- 2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [84 x]$

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 $n = 3$ .

$- 2 (3 x^{2}) + \frac{d}{d x} [84 x]$

Step 1.2.2.3

Multiply $3$ by $- 2$ .

$- 6 x^{2} + \frac{d}{d x} [84 x]$

Step 1.2.3

Evaluate $\frac{d}{d x} [84 x]$ .

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Step 1.2.3.1

Since $84$ is constant with respect to $x$ , the derivative of $84 x$ with respect to $x$ is $84 \frac{d}{d x} [x]$ .

$- 6 x^{2} + 84 \frac{d}{d x} [x]$

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 $n = 1$ .

$- 6 x^{2} + 84 \cdot 1$

Step 1.2.3.3

Multiply $84$ by $1$ .

$f'' (x) = - 6 x^{2} + 84$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- 6 x^{2} + 84$ .

$- 6 x^{2} + 84$

Step 2

Set the second derivative equal to $0$ then solve the equation $- 6 x^{2} + 84 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- 6 x^{2} + 84 = 0$

Step 2.2

Subtract $84$ from both sides of the equation.

$- 6 x^{2} = - 84$

Step 2.3

Divide each term in $- 6 x^{2} = - 84$ by $- 6$ and simplify.

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Step 2.3.1

Divide each term in $- 6 x^{2} = - 84$ by $- 6$ .

$\frac{- 6 x^{2}}{- 6} = \frac{- 84}{- 6}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 6$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 6 x^{2}}{- 6} = \frac{- 84}{- 6}$

Step 2.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 84}{- 6}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 84$ by $- 6$ .

$x^{2} = 14$

Step 2.4

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

$x = \pm \sqrt{14}$

Step 2.5

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

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Step 2.5.1

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

$x = \sqrt{14}$

Step 2.5.2

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

$x = - \sqrt{14}$

Step 2.5.3

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

$x = \sqrt{14}, - \sqrt{14}$

Step 3

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

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Step 3.1

Substitute $\sqrt{14}$ in $f (x) = - \frac{1}{2} x^{4} + 42 x^{2}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $\sqrt{14}$ in the expression.

$f (\sqrt{14}) = - \frac{1}{2} \cdot {(\sqrt{14})}^{4} + 42 {(\sqrt{14})}^{2}$

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

Rewrite ${\sqrt{14}}^{4}$ as $14^{2}$ .

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Step 3.1.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$f (\sqrt{14}) = - \frac{1}{2} \cdot {(14^{\frac{1}{2}})}^{4} + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{1}{2} \cdot 4} + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.1.3

Combine $\frac{1}{2}$ and $4$ .

$f (\sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{4}{2}} + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.1.4

Cancel the common factor of $4$ and $2$ .

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Step 3.1.2.1.1.4.1

Factor $2$ out of $4$ .

$f (\sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{2 \cdot 2}{2}} + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.1.4.2

Cancel the common factors.

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Step 3.1.2.1.1.4.2.1

Factor $2$ out of $2$ .

$f (\sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{2 \cdot 2}{2 (1)}} + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.1.4.2.2

Cancel the common factor.

$f (\sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{2 \cdot 2}{2 \cdot 1}} + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.1.4.2.3

Rewrite the expression.

$f (\sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{2}{1}} + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.1.4.2.4

Divide $2$ by $1$ .

$f (\sqrt{14}) = - \frac{1}{2} \cdot 14^{2} + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.2

Raise $14$ to the power of $2$ .

$f (\sqrt{14}) = - \frac{1}{2} \cdot 196 + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.3

Cancel the common factor of $2$ .

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Step 3.1.2.1.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (\sqrt{14}) = \frac{- 1}{2} \cdot 196 + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.3.2

Factor $2$ out of $196$ .

$f (\sqrt{14}) = \frac{- 1}{2} \cdot (2 (98)) + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.3.3

Cancel the common factor.

$f (\sqrt{14}) = \frac{- 1}{2} \cdot (2 \cdot 98) + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.3.4

Rewrite the expression.

$f (\sqrt{14}) = - 1 \cdot 98 + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.4

Multiply $- 1$ by $98$ .

$f (\sqrt{14}) = - 98 + 42 {(\sqrt{14})}^{2}$

Step 3.1.2.1.5

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

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Step 3.1.2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$f (\sqrt{14}) = - 98 + 42 {(14^{\frac{1}{2}})}^{2}$

Step 3.1.2.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\sqrt{14}) = - 98 + 42 \cdot 14^{\frac{1}{2} \cdot 2}$

Step 3.1.2.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$f (\sqrt{14}) = - 98 + 42 \cdot 14^{\frac{2}{2}}$

Step 3.1.2.1.5.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.5.4.1

Cancel the common factor.

$f (\sqrt{14}) = - 98 + 42 \cdot 14^{\frac{2}{2}}$

Step 3.1.2.1.5.4.2

Rewrite the expression.

$f (\sqrt{14}) = - 98 + 42 \cdot 14$

Step 3.1.2.1.5.5

Evaluate the exponent.

$f (\sqrt{14}) = - 98 + 42 \cdot 14$

Step 3.1.2.1.6

Multiply $42$ by $14$ .

$f (\sqrt{14}) = - 98 + 588$

Step 3.1.2.2

Add $- 98$ and $588$ .

$f (\sqrt{14}) = 490$

Step 3.1.2.3

The final answer is $490$ .

$490$

Step 3.2

The point found by substituting $\sqrt{14}$ in $f (x) = - \frac{1}{2} x^{4} + 42 x^{2}$ is $(\sqrt{14}, 490)$ . This point can be an inflection point.

$(\sqrt{14}, 490)$

Step 3.3

Substitute $- \sqrt{14}$ in $f (x) = - \frac{1}{2} x^{4} + 42 x^{2}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- \sqrt{14}$ in the expression.

$f (- \sqrt{14}) = - \frac{1}{2} \cdot {(- \sqrt{14})}^{4} + 42 {(- \sqrt{14})}^{2}$

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

Apply the product rule to $- \sqrt{14}$ .

$f (- \sqrt{14}) = - \frac{1}{2} \cdot ({(- 1)}^{4} {\sqrt{14}}^{4}) + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.2

Multiply $- 1$ by ${(- 1)}^{4}$ by adding the exponents.

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Step 3.3.2.1.2.1

Move ${(- 1)}^{4}$ .

$f (- \sqrt{14}) = {(- 1)}^{4} \cdot (- 1 (\frac{1}{2})) \cdot {\sqrt{14}}^{4} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.2.2

Multiply ${(- 1)}^{4}$ by $- 1$ .

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Step 3.3.2.1.2.2.1

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

$f (- \sqrt{14}) = {(- 1)}^{4} \cdot ((- 1) (\frac{1}{2})) \cdot {\sqrt{14}}^{4} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.2.2.2

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

$f (- \sqrt{14}) = {(- 1)}^{4 + 1} (\frac{1}{2}) \cdot {\sqrt{14}}^{4} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.2.3

Add $4$ and $1$ .

$f (- \sqrt{14}) = {(- 1)}^{5} (\frac{1}{2}) \cdot {\sqrt{14}}^{4} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.3

Raise $- 1$ to the power of $5$ .

$f (- \sqrt{14}) = - \frac{1}{2} \cdot {\sqrt{14}}^{4} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.4

Rewrite ${\sqrt{14}}^{4}$ as $14^{2}$ .

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Step 3.3.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$f (- \sqrt{14}) = - \frac{1}{2} \cdot {(14^{\frac{1}{2}})}^{4} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{1}{2} \cdot 4} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.4.3

Combine $\frac{1}{2}$ and $4$ .

$f (- \sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{4}{2}} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.4.4

Cancel the common factor of $4$ and $2$ .

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Step 3.3.2.1.4.4.1

Factor $2$ out of $4$ .

$f (- \sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{2 \cdot 2}{2}} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.4.4.2

Cancel the common factors.

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Step 3.3.2.1.4.4.2.1

Factor $2$ out of $2$ .

$f (- \sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{2 \cdot 2}{2 (1)}} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.4.4.2.2

Cancel the common factor.

$f (- \sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{2 \cdot 2}{2 \cdot 1}} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.4.4.2.3

Rewrite the expression.

$f (- \sqrt{14}) = - \frac{1}{2} \cdot 14^{\frac{2}{1}} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.4.4.2.4

Divide $2$ by $1$ .

$f (- \sqrt{14}) = - \frac{1}{2} \cdot 14^{2} + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.5

Raise $14$ to the power of $2$ .

$f (- \sqrt{14}) = - \frac{1}{2} \cdot 196 + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.6

Cancel the common factor of $2$ .

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Step 3.3.2.1.6.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (- \sqrt{14}) = \frac{- 1}{2} \cdot 196 + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.6.2

Factor $2$ out of $196$ .

$f (- \sqrt{14}) = \frac{- 1}{2} \cdot (2 (98)) + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.6.3

Cancel the common factor.

$f (- \sqrt{14}) = \frac{- 1}{2} \cdot (2 \cdot 98) + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.6.4

Rewrite the expression.

$f (- \sqrt{14}) = - 1 \cdot 98 + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.7

Multiply $- 1$ by $98$ .

$f (- \sqrt{14}) = - 98 + 42 {(- \sqrt{14})}^{2}$

Step 3.3.2.1.8

Apply the product rule to $- \sqrt{14}$ .

$f (- \sqrt{14}) = - 98 + 42 ({(- 1)}^{2} {\sqrt{14}}^{2})$

Step 3.3.2.1.9

Raise $- 1$ to the power of $2$ .

$f (- \sqrt{14}) = - 98 + 42 (1 {\sqrt{14}}^{2})$

Step 3.3.2.1.10

Multiply ${\sqrt{14}}^{2}$ by $1$ .

$f (- \sqrt{14}) = - 98 + 42 {\sqrt{14}}^{2}$

Step 3.3.2.1.11

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

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Step 3.3.2.1.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$f (- \sqrt{14}) = - 98 + 42 {(14^{\frac{1}{2}})}^{2}$

Step 3.3.2.1.11.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \sqrt{14}) = - 98 + 42 \cdot 14^{\frac{1}{2} \cdot 2}$

Step 3.3.2.1.11.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \sqrt{14}) = - 98 + 42 \cdot 14^{\frac{2}{2}}$

Step 3.3.2.1.11.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.11.4.1

Cancel the common factor.

$f (- \sqrt{14}) = - 98 + 42 \cdot 14^{\frac{2}{2}}$

Step 3.3.2.1.11.4.2

Rewrite the expression.

$f (- \sqrt{14}) = - 98 + 42 \cdot 14$

Step 3.3.2.1.11.5

Evaluate the exponent.

$f (- \sqrt{14}) = - 98 + 42 \cdot 14$

Step 3.3.2.1.12

Multiply $42$ by $14$ .

$f (- \sqrt{14}) = - 98 + 588$

Step 3.3.2.2

Add $- 98$ and $588$ .

$f (- \sqrt{14}) = 490$

Step 3.3.2.3

The final answer is $490$ .

$490$

Step 3.4

The point found by substituting $- \sqrt{14}$ in $f (x) = - \frac{1}{2} x^{4} + 42 x^{2}$ is $(- \sqrt{14}, 490)$ . This point can be an inflection point.

$(- \sqrt{14}, 490)$

Step 3.5

Determine the points that could be inflection points.

$(\sqrt{14}, 490), (- \sqrt{14}, 490)$

Step 4

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

$(- \infty, - \sqrt{14}) \cup (- \sqrt{14}, \sqrt{14}) \cup (\sqrt{14}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \sqrt{14})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 3.84165738$ in the expression.

$f'' (- 3.84165738) = - 6 {(- 3.84165738)}^{2} + 84$

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 $- 3.84165738$ to the power of $2$ .

$f'' (- 3.84165738) = - 6 \cdot 14.75833147 + 84$

Step 5.2.1.2

Multiply $- 6$ by $14.75833147$ .

$f'' (- 3.84165738) = - 88.54998886 + 84$

Step 5.2.2

Add $- 88.54998886$ and $84$ .

$f'' (- 3.84165738) = - 4.54998886$

Step 5.2.3

The final answer is $- 4.54998886$ .

$- 4.54998886$

Step 5.3

At $- 3.84165738$ , the second derivative is $- 4.54998886$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \sqrt{14})$

Decreasing on $(- \infty, - \sqrt{14})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- \sqrt{14}, \sqrt{14})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

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

$f'' (0) = - 6 {(0)}^{2} + 84$

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

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

$f'' (0) = - 6 \cdot 0 + 84$

Step 6.2.1.2

Multiply $- 6$ by $0$ .

$f'' (0) = 0 + 84$

Step 6.2.2

Add $0$ and $84$ .

$f'' (0) = 84$

Step 6.2.3

The final answer is $84$ .

$84$

Step 6.3

At $0$ , the second derivative is $84$ . Since this is positive, the second derivative is increasing on the interval $(- \sqrt{14}, \sqrt{14})$ .

Increasing on $(- \sqrt{14}, \sqrt{14})$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(\sqrt{14}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $3.84165738$ in the expression.

$f'' (3.84165738) = - 6 {(3.84165738)}^{2} + 84$

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 $3.84165738$ to the power of $2$ .

$f'' (3.84165738) = - 6 \cdot 14.75833147 + 84$

Step 7.2.1.2

Multiply $- 6$ by $14.75833147$ .

$f'' (3.84165738) = - 88.54998886 + 84$

Step 7.2.2

Add $- 88.54998886$ and $84$ .

$f'' (3.84165738) = - 4.54998886$

Step 7.2.3

The final answer is $- 4.54998886$ .

$- 4.54998886$

Step 7.3

At $3.84165738$ , the second derivative is $- 4.54998886$ . Since this is negative, the second derivative is decreasing on the interval $(\sqrt{14}, \infty)$

Decreasing on $(\sqrt{14}, \infty)$ 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 $(- \sqrt{14}, 490), (\sqrt{14}, 490)$ .

$(- \sqrt{14}, 490), (\sqrt{14}, 490)$

Step 9