Calculus Examples

$f (x) = \arctan (x^{2})$

Step 1

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

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

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Step 1.1.1.1.1

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

$f' (x) = \frac{d}{d u} (\arctan (u)) \frac{d}{d x} (x^{2})$

Step 1.1.1.1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

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

Step 1.1.1.1.3

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

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

Step 1.1.1.2

Differentiate using the Power Rule.

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Step 1.1.1.2.1

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

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Step 1.1.1.2.1.1

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

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

Step 1.1.1.2.1.2

Multiply $2$ by $2$ .

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

Combine fractions.

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Step 1.1.1.2.3.1

Combine $2$ and $\frac{1}{1 + x^{4}}$ .

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

Step 1.1.1.2.3.2

Combine $\frac{2}{1 + x^{4}}$ and $x$ .

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

Step 1.1.1.2.3.3

Reorder terms.

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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 $g (x) = x^{4} + 1$ .

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

Step 1.1.2.3

Differentiate.

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Step 1.1.2.3.1

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

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

Step 1.1.2.3.2

Multiply $x^{4} + 1$ by $1$ .

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

Step 1.1.2.3.3

By the Sum Rule, the derivative of $x^{4} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [1]$ .

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

Step 1.1.2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 1.1.2.3.5

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

$f'' (x) = 2 (\frac{x^{4} + 1 - x (4 x^{3} + 0)}{{(x^{4} + 1)}^{2}})$

Step 1.1.2.3.6

Simplify the expression.

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Step 1.1.2.3.6.1

Add $4 x^{3}$ and $0$ .

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

Step 1.1.2.3.6.2

Multiply $4$ by $- 1$ .

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

Step 1.1.2.4

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 1.1.2.4.1

Move $x^{3}$ .

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

Step 1.1.2.4.2

Multiply $x^{3}$ by $x$ .

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Step 1.1.2.4.2.1

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

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

Step 1.1.2.4.2.2

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

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

Step 1.1.2.4.3

Add $3$ and $1$ .

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

Step 1.1.2.5

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

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

Step 1.1.2.6

Combine $2$ and $\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}}$ .

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

Step 1.1.2.7

Simplify.

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Step 1.1.2.7.1

Apply the distributive property.

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

Step 1.1.2.7.2

Simplify each term.

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Step 1.1.2.7.2.1

Multiply $- 3$ by $2$ .

$f'' (x) = \frac{- 6 x^{4} + 2 \cdot 1}{{(x^{4} + 1)}^{2}}$

Step 1.1.2.7.2.2

Multiply $2$ by $1$ .

$f'' (x) = \frac{- 6 x^{4} + 2}{{(x^{4} + 1)}^{2}}$

Step 1.1.2.7.3

Factor $2$ out of $- 6 x^{4} + 2$ .

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Step 1.1.2.7.3.1

Factor $2$ out of $- 6 x^{4}$ .

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

Step 1.1.2.7.3.2

Factor $2$ out of $2$ .

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

Step 1.1.2.7.3.3

Factor $2$ out of $2 (- 3 x^{4}) + 2 (1)$ .

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

Step 1.1.2.7.4

Factor $- 1$ out of $- 3 x^{4}$ .

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

Step 1.1.2.7.5

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

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

Step 1.1.2.7.6

Factor $- 1$ out of $- (3 x^{4}) - 1 (- 1)$ .

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

Step 1.1.2.7.7

Rewrite $- (3 x^{4} - 1)$ as $- 1 (3 x^{4} - 1)$ .

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

Step 1.1.2.7.8

Move the negative in front of the fraction.

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

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{2 (3 x^{4} - 1)}{{(x^{4} + 1)}^{2}}$ .

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

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- \frac{2 (3 x^{4} - 1)}{{(x^{4} + 1)}^{2}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- \frac{2 (3 x^{4} - 1)}{{(x^{4} + 1)}^{2}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$2 (3 x^{4} - 1) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Divide each term in $2 (3 x^{4} - 1) = 0$ by $2$ and simplify.

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Step 1.2.3.1.1

Divide each term in $2 (3 x^{4} - 1) = 0$ by $2$ .

$\frac{2 (3 x^{4} - 1)}{2} = \frac{0}{2}$

Step 1.2.3.1.2

Simplify the left side.

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Step 1.2.3.1.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.1.2.1.1

Cancel the common factor.

$\frac{2 (3 x^{4} - 1)}{2} = \frac{0}{2}$

Step 1.2.3.1.2.1.2

Divide $3 x^{4} - 1$ by $1$ .

$3 x^{4} - 1 = \frac{0}{2}$

Step 1.2.3.1.3

Simplify the right side.

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Step 1.2.3.1.3.1

Divide $0$ by $2$ .

$3 x^{4} - 1 = 0$

Step 1.2.3.2

Add $1$ to both sides of the equation.

$3 x^{4} = 1$

Step 1.2.3.3

Divide each term in $3 x^{4} = 1$ by $3$ and simplify.

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Step 1.2.3.3.1

Divide each term in $3 x^{4} = 1$ by $3$ .

$\frac{3 x^{4}}{3} = \frac{1}{3}$

Step 1.2.3.3.2

Simplify the left side.

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Step 1.2.3.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.3.2.1.1

Cancel the common factor.

$\frac{3 x^{4}}{3} = \frac{1}{3}$

Step 1.2.3.3.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{1}{3}$

Step 1.2.3.4

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

$x = \pm \sqrt[4]{\frac{1}{3}}$

Step 1.2.3.5

Simplify $\pm \sqrt[4]{\frac{1}{3}}$ .

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Step 1.2.3.5.1

Rewrite $\sqrt[4]{\frac{1}{3}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{3}}$ .

$x = \pm \frac{\sqrt[4]{1}}{\sqrt[4]{3}}$

Step 1.2.3.5.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt[4]{3}}$

Step 1.2.3.5.3

Multiply $\frac{1}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$x = \pm \frac{1}{\sqrt[4]{3}} \cdot \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$

Step 1.2.3.5.4

Combine and simplify the denominator.

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Step 1.2.3.5.4.1

Multiply $\frac{1}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{\sqrt[4]{3} {\sqrt[4]{3}}^{3}}$

Step 1.2.3.5.4.2

Raise $\sqrt[4]{3}$ to the power of $1$ .

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1} {\sqrt[4]{3}}^{3}}$

Step 1.2.3.5.4.3

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

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1 + 3}}$

Step 1.2.3.5.4.4

Add $1$ and $3$ .

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{4}}$

Step 1.2.3.5.4.5

Rewrite ${\sqrt[4]{3}}^{4}$ as $3$ .

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Step 1.2.3.5.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{{(3^{\frac{1}{4}})}^{4}}$

Step 1.2.3.5.4.5.2

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

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{\frac{1}{4} \cdot 4}}$

Step 1.2.3.5.4.5.3

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

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 1.2.3.5.4.5.4

Cancel the common factor of $4$ .

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Step 1.2.3.5.4.5.4.1

Cancel the common factor.

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 1.2.3.5.4.5.4.2

Rewrite the expression.

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{1}}$

Step 1.2.3.5.4.5.5

Evaluate the exponent.

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{3}$

Step 1.2.3.5.5

Simplify the numerator.

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Step 1.2.3.5.5.1

Rewrite ${\sqrt[4]{3}}^{3}$ as $\sqrt[4]{3^{3}}$ .

$x = \pm \frac{\sqrt[4]{3^{3}}}{3}$

Step 1.2.3.5.5.2

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

$x = \pm \frac{\sqrt[4]{27}}{3}$

Step 1.2.3.6

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

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Step 1.2.3.6.1

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

$x = \frac{\sqrt[4]{27}}{3}$

Step 1.2.3.6.2

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

$x = - \frac{\sqrt[4]{27}}{3}$

Step 1.2.3.6.3

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

$x = \frac{\sqrt[4]{27}}{3}, - \frac{\sqrt[4]{27}}{3}$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

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

$(- \infty, - \frac{\sqrt[4]{27}}{3}) \cup (- \frac{\sqrt[4]{27}}{3}, \frac{\sqrt[4]{27}}{3}) \cup (\frac{\sqrt[4]{27}}{3}, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - \frac{\sqrt[4]{27}}{3})$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

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

$f'' (- 3) = - \frac{2 (3 \cdot 81 - 1)}{{({(- 3)}^{4} + 1)}^{2}}$

Step 4.2.1.2

Multiply $3$ by $81$ .

$f'' (- 3) = - \frac{2 (243 - 1)}{{({(- 3)}^{4} + 1)}^{2}}$

Step 4.2.1.3

Subtract $1$ from $243$ .

$f'' (- 3) = - \frac{2 \cdot 242}{{({(- 3)}^{4} + 1)}^{2}}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

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

$f'' (- 3) = - \frac{2 \cdot 242}{{(81 + 1)}^{2}}$

Step 4.2.2.2

Add $81$ and $1$ .

$f'' (- 3) = - \frac{2 \cdot 242}{82^{2}}$

Step 4.2.2.3

Raise $82$ to the power of $2$ .

$f'' (- 3) = - \frac{2 \cdot 242}{6724}$

Step 4.2.3

Reduce the expression by cancelling the common factors.

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Step 4.2.3.1

Multiply $2$ by $242$ .

$f'' (- 3) = - \frac{484}{6724}$

Step 4.2.3.2

Cancel the common factor of $484$ and $6724$ .

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Step 4.2.3.2.1

Factor $4$ out of $484$ .

$f'' (- 3) = - \frac{4 (121)}{6724}$

Step 4.2.3.2.2

Cancel the common factors.

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Step 4.2.3.2.2.1

Factor $4$ out of $6724$ .

$f'' (- 3) = - \frac{4 \cdot 121}{4 \cdot 1681}$

Step 4.2.3.2.2.2

Cancel the common factor.

$f'' (- 3) = - \frac{4 \cdot 121}{4 \cdot 1681}$

Step 4.2.3.2.2.3

Rewrite the expression.

$f'' (- 3) = - \frac{121}{1681}$

Step 4.2.4

The final answer is $- \frac{121}{1681}$ .

$- \frac{121}{1681}$

Step 4.3

The graph is concave down on the interval $(- \infty, - \frac{\sqrt[4]{27}}{3})$ because $f'' (- 3)$ is negative.

Concave down on $(- \infty, - \frac{\sqrt[4]{27}}{3})$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(- \frac{\sqrt[4]{27}}{3}, \frac{\sqrt[4]{27}}{3})$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (0) = - \frac{2 (3 {(0)}^{4} - 1)}{{({(0)}^{4} + 1)}^{2}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

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

$f'' (0) = - \frac{2 (3 \cdot 0 - 1)}{{(0^{4} + 1)}^{2}}$

Step 5.2.1.2

Multiply $3$ by $0$ .

$f'' (0) = - \frac{2 (0 - 1)}{{(0^{4} + 1)}^{2}}$

Step 5.2.1.3

Subtract $1$ from $0$ .

$f'' (0) = - \frac{2 \cdot - 1}{{(0^{4} + 1)}^{2}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

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

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

Step 5.2.2.2

Add $0$ and $1$ .

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

Step 5.2.2.3

One to any power is one.

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

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Multiply $2$ by $- 1$ .

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

Step 5.2.3.2

Divide $- 2$ by $1$ .

$f'' (0) = 2$

Step 5.2.3.3

Multiply $- 1$ by $- 2$ .

$f'' (0) = 2$

Step 5.2.4

The final answer is $2$ .

$2$

Step 5.3

The graph is concave up on the interval $(- \frac{\sqrt[4]{27}}{3}, \frac{\sqrt[4]{27}}{3})$ because $f'' (0)$ is positive.

Concave up on $(- \frac{\sqrt[4]{27}}{3}, \frac{\sqrt[4]{27}}{3})$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(\frac{\sqrt[4]{27}}{3}, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

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

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $3$ by ${(3)}^{4}$ by adding the exponents.

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Step 6.2.1.1

Multiply $3$ by ${(3)}^{4}$ .

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Step 6.2.1.1.1

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

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

Step 6.2.1.1.2

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

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

Step 6.2.1.2

Add $1$ and $4$ .

$f'' (3) = - \frac{2 (3^{5} - 1)}{{({(3)}^{4} + 1)}^{2}}$

Step 6.2.2

Simplify the numerator.

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Step 6.2.2.1

Raise $3$ to the power of $5$ .

$f'' (3) = - \frac{2 (243 - 1)}{{(3^{4} + 1)}^{2}}$

Step 6.2.2.2

Subtract $1$ from $243$ .

$f'' (3) = - \frac{2 \cdot 242}{{(3^{4} + 1)}^{2}}$

Step 6.2.3

Simplify the denominator.

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Step 6.2.3.1

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

$f'' (3) = - \frac{2 \cdot 242}{{(81 + 1)}^{2}}$

Step 6.2.3.2

Add $81$ and $1$ .

$f'' (3) = - \frac{2 \cdot 242}{82^{2}}$

Step 6.2.3.3

Raise $82$ to the power of $2$ .

$f'' (3) = - \frac{2 \cdot 242}{6724}$

Step 6.2.4

Reduce the expression by cancelling the common factors.

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Step 6.2.4.1

Multiply $2$ by $242$ .

$f'' (3) = - \frac{484}{6724}$

Step 6.2.4.2

Cancel the common factor of $484$ and $6724$ .

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Step 6.2.4.2.1

Factor $4$ out of $484$ .

$f'' (3) = - \frac{4 (121)}{6724}$

Step 6.2.4.2.2

Cancel the common factors.

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Step 6.2.4.2.2.1

Factor $4$ out of $6724$ .

$f'' (3) = - \frac{4 \cdot 121}{4 \cdot 1681}$

Step 6.2.4.2.2.2

Cancel the common factor.

$f'' (3) = - \frac{4 \cdot 121}{4 \cdot 1681}$

Step 6.2.4.2.2.3

Rewrite the expression.

$f'' (3) = - \frac{121}{1681}$

Step 6.2.5

The final answer is $- \frac{121}{1681}$ .

$- \frac{121}{1681}$

Step 6.3

The graph is concave down on the interval $(\frac{\sqrt[4]{27}}{3}, \infty)$ because $f'' (3)$ is negative.

Concave down on $(\frac{\sqrt[4]{27}}{3}, \infty)$ since $f'' (x)$ is negative

Step 7

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

Concave down on $(- \infty, - \frac{\sqrt[4]{27}}{3})$ since $f'' (x)$ is negative

Concave up on $(- \frac{\sqrt[4]{27}}{3}, \frac{\sqrt[4]{27}}{3})$ since $f'' (x)$ is positive

Concave down on $(\frac{\sqrt[4]{27}}{3}, \infty)$ since $f'' (x)$ is negative

Step 8