Calculus Examples

$y = \frac{1}{x^{2} + 1}$

Step 1

Write $y = \frac{1}{x^{2} + 1}$ as a function.

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

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

Tap for more steps...

Step 2.1

Find the second derivative.

Tap for more steps...

Step 2.1.1

Find the first derivative.

Tap for more steps...

Step 2.1.1.1

Rewrite $\frac{1}{x^{2} + 1}$ as ${(x^{2} + 1)}^{- 1}$ .

$f' (x) = \frac{d}{d x} ({(x^{2} + 1)}^{- 1})$

Step 2.1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} + 1$ .

Tap for more steps...

Step 2.1.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$f' (x) = \frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2} + 1)$

Step 2.1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 1$ .

$f' (x) = - u^{- 2} \frac{d}{d x} (x^{2} + 1)$

Step 2.1.1.2.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

$f' (x) = - {(x^{2} + 1)}^{- 2} \frac{d}{d x} (x^{2} + 1)$

Step 2.1.1.3

Differentiate.

Tap for more steps...

Step 2.1.1.3.1

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$f' (x) = - {(x^{2} + 1)}^{- 2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (1))$

Step 2.1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = - {(x^{2} + 1)}^{- 2} (2 x + \frac{d}{d x} (1))$

Step 2.1.1.3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f' (x) = - {(x^{2} + 1)}^{- 2} (2 x + 0)$

Step 2.1.1.3.4

Simplify the expression.

Tap for more steps...

Step 2.1.1.3.4.1

Add $2 x$ and $0$ .

$f' (x) = - {(x^{2} + 1)}^{- 2} (2 x)$

Step 2.1.1.3.4.2

Multiply $2$ by $- 1$ .

$f' (x) = - 2 {(x^{2} + 1)}^{- 2} x$

Step 2.1.1.4

Simplify.

Tap for more steps...

Step 2.1.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = - 2 \frac{1}{{(x^{2} + 1)}^{2}} \cdot x$

Step 2.1.1.4.2

Combine terms.

Tap for more steps...

Step 2.1.1.4.2.1

Combine $- 2$ and $\frac{1}{{(x^{2} + 1)}^{2}}$ .

$f' (x) = \frac{- 2}{{(x^{2} + 1)}^{2}} \cdot x$

Step 2.1.1.4.2.2

Move the negative in front of the fraction.

$f' (x) = - \frac{2}{{(x^{2} + 1)}^{2}} \cdot x$

Step 2.1.1.4.2.3

Combine $x$ and $\frac{2}{{(x^{2} + 1)}^{2}}$ .

$f' (x) = - \frac{x \cdot 2}{{(x^{2} + 1)}^{2}}$

Step 2.1.1.4.2.4

Move $2$ to the left of $x$ .

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

Step 2.1.2

Find the second derivative.

Tap for more steps...

Step 2.1.2.1

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

$f'' (x) = - 2 \frac{d}{d x} \frac{x}{{(x^{2} + 1)}^{2}}$

Step 2.1.2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = {(x^{2} + 1)}^{2}$ .

$f'' (x) = - 2 \frac{{(x^{2} + 1)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 1)}^{2}}{{({(x^{2} + 1)}^{2})}^{2}}$

Step 2.1.2.3

Differentiate using the Power Rule.

Tap for more steps...

Step 2.1.2.3.1

Multiply the exponents in ${({(x^{2} + 1)}^{2})}^{2}$ .

Tap for more steps...

Step 2.1.2.3.1.1

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

$f'' (x) = - 2 \frac{{(x^{2} + 1)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 1)}^{2}}{{(x^{2} + 1)}^{2 \cdot 2}}$

Step 2.1.2.3.1.2

Multiply $2$ by $2$ .

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

Step 2.1.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

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

Step 2.1.2.3.3

Multiply ${(x^{2} + 1)}^{2}$ by $1$ .

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

Step 2.1.2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 1$ .

Tap for more steps...

Step 2.1.2.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

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

Step 2.1.2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 2.1.2.4.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 2.1.2.5

Simplify with factoring out.

Tap for more steps...

Step 2.1.2.5.1

Multiply $2$ by $- 1$ .

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

Step 2.1.2.5.2

Factor $x^{2} + 1$ out of ${(x^{2} + 1)}^{2} - 2 x ((x^{2} + 1) \frac{d}{d x} [x^{2} + 1])$ .

Tap for more steps...

Step 2.1.2.5.2.1

Factor $x^{2} + 1$ out of ${(x^{2} + 1)}^{2}$ .

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

Step 2.1.2.5.2.2

Factor $x^{2} + 1$ out of $- 2 x ((x^{2} + 1) \frac{d}{d x} [x^{2} + 1])$ .

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

Step 2.1.2.5.2.3

Factor $x^{2} + 1$ out of $(x^{2} + 1) (x^{2} + 1) + (x^{2} + 1) (- 2 x (\frac{d}{d x} [x^{2} + 1]))$ .

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

Step 2.1.2.6

Cancel the common factors.

Tap for more steps...

Step 2.1.2.6.1

Factor $x^{2} + 1$ out of ${(x^{2} + 1)}^{4}$ .

$f'' (x) = - 2 \frac{(x^{2} + 1) (x^{2} + 1 - 2 x (\frac{d}{d x} (x^{2} + 1)))}{(x^{2} + 1) {(x^{2} + 1)}^{3}}$

Step 2.1.2.6.2

Cancel the common factor.

$f'' (x) = - 2 \frac{(x^{2} + 1) (x^{2} + 1 - 2 x (\frac{d}{d x} (x^{2} + 1)))}{(x^{2} + 1) {(x^{2} + 1)}^{3}}$

Step 2.1.2.6.3

Rewrite the expression.

$f'' (x) = - 2 \frac{x^{2} + 1 - 2 x (\frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.7

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$f'' (x) = - 2 \frac{x^{2} + 1 - 2 x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = - 2 \frac{x^{2} + 1 - 2 x (2 x + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.9

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = - 2 \frac{x^{2} + 1 - 2 x (2 x + 0)}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.10

Simplify the expression.

Tap for more steps...

Step 2.1.2.10.1

Add $2 x$ and $0$ .

$f'' (x) = - 2 \frac{x^{2} + 1 - 2 x (2 x)}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.10.2

Multiply $2$ by $- 2$ .

$f'' (x) = - 2 \frac{x^{2} + 1 - 4 x \cdot x}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.11

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

$f'' (x) = - 2 \frac{x^{2} + 1 - 4 (x \cdot x)}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.12

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

$f'' (x) = - 2 \frac{x^{2} + 1 - 4 (x \cdot x)}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.13

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

$f'' (x) = - 2 \frac{x^{2} + 1 - 4 x^{1 + 1}}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.14

Add $1$ and $1$ .

$f'' (x) = - 2 \frac{x^{2} + 1 - 4 x^{2}}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.15

Subtract $4 x^{2}$ from $x^{2}$ .

$f'' (x) = - 2 \frac{- 3 x^{2} + 1}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.16

Combine $- 2$ and $\frac{- 3 x^{2} + 1}{{(x^{2} + 1)}^{3}}$ .

$f'' (x) = \frac{- 2 (- 3 x^{2} + 1)}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.17

Move the negative in front of the fraction.

$f'' (x) = - \frac{(2) (- 3 x^{2} + 1)}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18

Simplify.

Tap for more steps...

Step 2.1.2.18.1

Apply the distributive property.

$f'' (x) = - \frac{2 (- 3 x^{2}) + 2 \cdot 1}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18.2

Simplify each term.

Tap for more steps...

Step 2.1.2.18.2.1

Multiply $- 3$ by $2$ .

$f'' (x) = - \frac{- 6 x^{2} + 2 \cdot 1}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18.2.2

Multiply $2$ by $1$ .

$f'' (x) = - \frac{- 6 x^{2} + 2}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18.3

Factor $2$ out of $- 6 x^{2} + 2$ .

Tap for more steps...

Step 2.1.2.18.3.1

Factor $2$ out of $- 6 x^{2}$ .

$f'' (x) = - \frac{2 (- 3 x^{2}) + 2}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18.3.2

Factor $2$ out of $2$ .

$f'' (x) = - \frac{2 (- 3 x^{2}) + 2 (1)}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18.3.3

Factor $2$ out of $2 (- 3 x^{2}) + 2 (1)$ .

$f'' (x) = - \frac{2 (- 3 x^{2} + 1)}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18.4

Factor $- 1$ out of $- 3 x^{2}$ .

$f'' (x) = - \frac{2 (- (3 x^{2}) + 1)}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18.5

Rewrite $1$ as $- 1 (- 1)$ .

$f'' (x) = - \frac{2 (- (3 x^{2}) - 1 \cdot - 1)}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18.6

Factor $- 1$ out of $- (3 x^{2}) - 1 (- 1)$ .

$f'' (x) = - \frac{2 (- (3 x^{2} - 1))}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18.7

Rewrite $- (3 x^{2} - 1)$ as $- 1 (3 x^{2} - 1)$ .

$f'' (x) = - \frac{2 (- 1 (3 x^{2} - 1))}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18.8

Move the negative in front of the fraction.

$f'' (x) = \frac{2 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}}$

Step 2.1.2.18.9

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{2 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}})$

Step 2.1.2.18.10

Multiply $\frac{2 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}}$ by $1$ .

$f'' (x) = \frac{2 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}}$ .

$\frac{2 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{2 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}} = 0$ .

Tap for more steps...

Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{2 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$2 (3 x^{2} - 1) = 0$

Step 2.2.3

Solve the equation for $x$ .

Tap for more steps...

Step 2.2.3.1

Divide each term in $2 (3 x^{2} - 1) = 0$ by $2$ and simplify.

Tap for more steps...

Step 2.2.3.1.1

Divide each term in $2 (3 x^{2} - 1) = 0$ by $2$ .

$\frac{2 (3 x^{2} - 1)}{2} = \frac{0}{2}$

Step 2.2.3.1.2

Simplify the left side.

Tap for more steps...

Step 2.2.3.1.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.3.1.2.1.1

Cancel the common factor.

$\frac{2 (3 x^{2} - 1)}{2} = \frac{0}{2}$

Step 2.2.3.1.2.1.2

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

$3 x^{2} - 1 = \frac{0}{2}$

Step 2.2.3.1.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1.3.1

Divide $0$ by $2$ .

$3 x^{2} - 1 = 0$

Step 2.2.3.2

Add $1$ to both sides of the equation.

$3 x^{2} = 1$

Step 2.2.3.3

Divide each term in $3 x^{2} = 1$ by $3$ and simplify.

Tap for more steps...

Step 2.2.3.3.1

Divide each term in $3 x^{2} = 1$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{1}{3}$

Step 2.2.3.3.2

Simplify the left side.

Tap for more steps...

Step 2.2.3.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.3.3.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{1}{3}$

Step 2.2.3.3.2.1.2

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

$x^{2} = \frac{1}{3}$

Step 2.2.3.4

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

$x = \pm \sqrt{\frac{1}{3}}$

Step 2.2.3.5

Simplify $\pm \sqrt{\frac{1}{3}}$ .

Tap for more steps...

Step 2.2.3.5.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{3}}$

Step 2.2.3.5.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{3}}$

Step 2.2.3.5.3

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

$x = \pm \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.2.3.5.4

Combine and simplify the denominator.

Tap for more steps...

Step 2.2.3.5.4.1

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

$x = \pm \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.2.3.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.2.3.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.2.3.5.4.4

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

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.2.3.5.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.2.3.5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 2.2.3.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.2.3.5.4.6.2

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

$x = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.2.3.5.4.6.3

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

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.2.3.5.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.3.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.2.3.5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{3}}{3^{1}}$

Step 2.2.3.5.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{3}}{3}$

Step 2.2.3.6

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

Tap for more steps...

Step 2.2.3.6.1

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

$x = \frac{\sqrt{3}}{3}$

Step 2.2.3.6.2

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

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

Step 2.2.3.6.3

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

$x = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Step 3

Find the domain of $f (x) = \frac{1}{x^{2} + 1}$ .

Tap for more steps...

Step 3.1

Set the denominator in $\frac{1}{x^{2} + 1}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 1 = 0$

Step 3.2

Solve for $x$ .

Tap for more steps...

Step 3.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 3.2.2

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

$x = \pm \sqrt{- 1}$

Step 3.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 3.2.4

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

Tap for more steps...

Step 3.2.4.1

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

$x = i$

Step 3.2.4.2

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

$x = - i$

Step 3.2.4.3

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

$x = i, - i$

Step 3.3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

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

Step 5

Substitute any number from the interval $(- \infty, - \frac{\sqrt{3}}{3})$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $- 3$ in the expression.

$f'' (- 3) = \frac{2 (3 {(- 3)}^{2} - 1)}{{({(- 3)}^{2} + 1)}^{3}}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify the numerator.

Tap for more steps...

Step 5.2.1.1

Raise $- 3$ to the power of $2$ .

$f'' (- 3) = \frac{2 (3 \cdot 9 - 1)}{{({(- 3)}^{2} + 1)}^{3}}$

Step 5.2.1.2

Multiply $3$ by $9$ .

$f'' (- 3) = \frac{2 (27 - 1)}{{({(- 3)}^{2} + 1)}^{3}}$

Step 5.2.1.3

Subtract $1$ from $27$ .

$f'' (- 3) = \frac{2 \cdot 26}{{({(- 3)}^{2} + 1)}^{3}}$

Step 5.2.2

Simplify the denominator.

Tap for more steps...

Step 5.2.2.1

Raise $- 3$ to the power of $2$ .

$f'' (- 3) = \frac{2 \cdot 26}{{(9 + 1)}^{3}}$

Step 5.2.2.2

Add $9$ and $1$ .

$f'' (- 3) = \frac{2 \cdot 26}{10^{3}}$

Step 5.2.2.3

Raise $10$ to the power of $3$ .

$f'' (- 3) = \frac{2 \cdot 26}{1000}$

Step 5.2.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.2.3.1

Multiply $2$ by $26$ .

$f'' (- 3) = \frac{52}{1000}$

Step 5.2.3.2

Cancel the common factor of $52$ and $1000$ .

Tap for more steps...

Step 5.2.3.2.1

Factor $4$ out of $52$ .

$f'' (- 3) = \frac{4 (13)}{1000}$

Step 5.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.2.2.1

Factor $4$ out of $1000$ .

$f'' (- 3) = \frac{4 \cdot 13}{4 \cdot 250}$

Step 5.2.3.2.2.2

Cancel the common factor.

$f'' (- 3) = \frac{4 \cdot 13}{4 \cdot 250}$

Step 5.2.3.2.2.3

Rewrite the expression.

$f'' (- 3) = \frac{13}{250}$

Step 5.2.4

The final answer is $\frac{13}{250}$ .

$\frac{13}{250}$

Step 5.3

The graph is concave up on the interval $(- \infty, - \frac{\sqrt{3}}{3})$ because $f'' (- 3)$ is positive.

Concave up on $(- \infty, - \frac{\sqrt{3}}{3})$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 6.1

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

$f'' (0) = \frac{2 (3 {(0)}^{2} - 1)}{{({(0)}^{2} + 1)}^{3}}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify the numerator.

Tap for more steps...

Step 6.2.1.1

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

$f'' (0) = \frac{2 (3 \cdot 0 - 1)}{{(0^{2} + 1)}^{3}}$

Step 6.2.1.2

Multiply $3$ by $0$ .

$f'' (0) = \frac{2 (0 - 1)}{{(0^{2} + 1)}^{3}}$

Step 6.2.1.3

Subtract $1$ from $0$ .

$f'' (0) = \frac{2 \cdot - 1}{{(0^{2} + 1)}^{3}}$

Step 6.2.2

Simplify the denominator.

Tap for more steps...

Step 6.2.2.1

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

$f'' (0) = \frac{2 \cdot - 1}{{(0 + 1)}^{3}}$

Step 6.2.2.2

Add $0$ and $1$ .

$f'' (0) = \frac{2 \cdot - 1}{1^{3}}$

Step 6.2.2.3

One to any power is one.

$f'' (0) = \frac{2 \cdot - 1}{1}$

Step 6.2.3

Simplify the expression.

Tap for more steps...

Step 6.2.3.1

Multiply $2$ by $- 1$ .

$f'' (0) = \frac{- 2}{1}$

Step 6.2.3.2

Divide $- 2$ by $1$ .

$f'' (0) = - 2$

Step 6.2.4

The final answer is $- 2$ .

$- 2$

Step 6.3

The graph is concave down on the interval $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ because $f'' (0)$ is negative.

Concave down on $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(\frac{\sqrt{3}}{3}, \infty)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $3$ in the expression.

$f'' (3) = \frac{2 (3 {(3)}^{2} - 1)}{{({(3)}^{2} + 1)}^{3}}$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

Tap for more steps...

Step 7.2.1.1

Multiply $3$ by ${(3)}^{2}$ .

Tap for more steps...

Step 7.2.1.1.1

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

$f'' (3) = \frac{2 (3 {(3)}^{2} - 1)}{{({(3)}^{2} + 1)}^{3}}$

Step 7.2.1.1.2

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

$f'' (3) = \frac{2 (3^{1 + 2} - 1)}{{({(3)}^{2} + 1)}^{3}}$

Step 7.2.1.2

Add $1$ and $2$ .

$f'' (3) = \frac{2 (3^{3} - 1)}{{({(3)}^{2} + 1)}^{3}}$

Step 7.2.2

Simplify the numerator.

Tap for more steps...

Step 7.2.2.1

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

$f'' (3) = \frac{2 (27 - 1)}{{(3^{2} + 1)}^{3}}$

Step 7.2.2.2

Subtract $1$ from $27$ .

$f'' (3) = \frac{2 \cdot 26}{{(3^{2} + 1)}^{3}}$

Step 7.2.3

Simplify the denominator.

Tap for more steps...

Step 7.2.3.1

Raise $3$ to the power of $2$ .

$f'' (3) = \frac{2 \cdot 26}{{(9 + 1)}^{3}}$

Step 7.2.3.2

Add $9$ and $1$ .

$f'' (3) = \frac{2 \cdot 26}{10^{3}}$

Step 7.2.3.3

Raise $10$ to the power of $3$ .

$f'' (3) = \frac{2 \cdot 26}{1000}$

Step 7.2.4

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 7.2.4.1

Multiply $2$ by $26$ .

$f'' (3) = \frac{52}{1000}$

Step 7.2.4.2

Cancel the common factor of $52$ and $1000$ .

Tap for more steps...

Step 7.2.4.2.1

Factor $4$ out of $52$ .

$f'' (3) = \frac{4 (13)}{1000}$

Step 7.2.4.2.2

Cancel the common factors.

Tap for more steps...

Step 7.2.4.2.2.1

Factor $4$ out of $1000$ .

$f'' (3) = \frac{4 \cdot 13}{4 \cdot 250}$

Step 7.2.4.2.2.2

Cancel the common factor.

$f'' (3) = \frac{4 \cdot 13}{4 \cdot 250}$

Step 7.2.4.2.2.3

Rewrite the expression.

$f'' (3) = \frac{13}{250}$

Step 7.2.5

The final answer is $\frac{13}{250}$ .

$\frac{13}{250}$

Step 7.3

The graph is concave up on the interval $(\frac{\sqrt{3}}{3}, \infty)$ because $f'' (3)$ is positive.

Concave up on $(\frac{\sqrt{3}}{3}, \infty)$ since $f'' (x)$ is positive

Step 8

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, - \frac{\sqrt{3}}{3})$ since $f'' (x)$ is positive

Concave down on $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ since $f'' (x)$ is negative

Concave up on $(\frac{\sqrt{3}}{3}, \infty)$ since $f'' (x)$ is positive

Step 9