Calculus Examples

$\ln (x^{4} + 27)$

Step 1

Write $\ln (x^{4} + 27)$ as a function.

$f (x) = \ln (x^{4} + 27)$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{4} + 27$ .

Tap for more steps...

Step 2.1.1.1

To apply the Chain Rule, set $u$ as $x^{4} + 27$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{4} + 27]$

Step 2.1.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [x^{4} + 27]$

Step 2.1.1.3

Replace all occurrences of $u$ with $x^{4} + 27$ .

$\frac{1}{x^{4} + 27} \frac{d}{d x} [x^{4} + 27]$

Step 2.1.2

Differentiate.

Tap for more steps...

Step 2.1.2.1

By the Sum Rule, the derivative of $x^{4} + 27$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [27]$ .

$\frac{1}{x^{4} + 27} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [27])$

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 $n = 4$ .

$\frac{1}{x^{4} + 27} (4 x^{3} + \frac{d}{d x} [27])$

Step 2.1.2.3

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$\frac{1}{x^{4} + 27} (4 x^{3} + 0)$

Step 2.1.2.4

Combine fractions.

Tap for more steps...

Step 2.1.2.4.1

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

$\frac{1}{x^{4} + 27} (4 x^{3})$

Step 2.1.2.4.2

Combine $4$ and $\frac{1}{x^{4} + 27}$ .

$\frac{4}{x^{4} + 27} x^{3}$

Step 2.1.2.4.3

Combine $\frac{4}{x^{4} + 27}$ and $x^{3}$ .

$f' (x) = \frac{4 x^{3}}{x^{4} + 27}$

Step 2.2

Find the second derivative.

Tap for more steps...

Step 2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 x^{3}}{x^{4} + 27}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x^{3}}{x^{4} + 27}]$ .

$4 \frac{d}{d x} [\frac{x^{3}}{x^{4} + 27}]$

Step 2.2.2

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 $f (x) = x^{3}$ and $g (x) = x^{4} + 27$ .

$4 \frac{(x^{4} + 27) \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [x^{4} + 27]}{{(x^{4} + 27)}^{2}}$

Step 2.2.3

Differentiate.

Tap for more steps...

Step 2.2.3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 \frac{(x^{4} + 27) (3 x^{2}) - x^{3} \frac{d}{d x} [x^{4} + 27]}{{(x^{4} + 27)}^{2}}$

Step 2.2.3.2

Move $3$ to the left of $x^{4} + 27$ .

$4 \frac{3 \cdot (x^{4} + 27) x^{2} - x^{3} \frac{d}{d x} [x^{4} + 27]}{{(x^{4} + 27)}^{2}}$

Step 2.2.3.3

By the Sum Rule, the derivative of $x^{4} + 27$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [27]$ .

$4 \frac{3 (x^{4} + 27) x^{2} - x^{3} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [27])}{{(x^{4} + 27)}^{2}}$

Step 2.2.3.4

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

$4 \frac{3 (x^{4} + 27) x^{2} - x^{3} (4 x^{3} + \frac{d}{d x} [27])}{{(x^{4} + 27)}^{2}}$

Step 2.2.3.5

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$4 \frac{3 (x^{4} + 27) x^{2} - x^{3} (4 x^{3} + 0)}{{(x^{4} + 27)}^{2}}$

Step 2.2.3.6

Simplify the expression.

Tap for more steps...

Step 2.2.3.6.1

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

$4 \frac{3 (x^{4} + 27) x^{2} - x^{3} (4 x^{3})}{{(x^{4} + 27)}^{2}}$

Step 2.2.3.6.2

Multiply $4$ by $- 1$ .

$4 \frac{3 (x^{4} + 27) x^{2} - 4 x^{3} x^{3}}{{(x^{4} + 27)}^{2}}$

Step 2.2.4

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

Tap for more steps...

Step 2.2.4.1

Move $x^{3}$ .

$4 \frac{3 (x^{4} + 27) x^{2} - 4 (x^{3} x^{3})}{{(x^{4} + 27)}^{2}}$

Step 2.2.4.2

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

$4 \frac{3 (x^{4} + 27) x^{2} - 4 x^{3 + 3}}{{(x^{4} + 27)}^{2}}$

Step 2.2.4.3

Add $3$ and $3$ .

$4 \frac{3 (x^{4} + 27) x^{2} - 4 x^{6}}{{(x^{4} + 27)}^{2}}$

Step 2.2.5

Combine $4$ and $\frac{3 (x^{4} + 27) x^{2} - 4 x^{6}}{{(x^{4} + 27)}^{2}}$ .

$\frac{4 (3 (x^{4} + 27) x^{2} - 4 x^{6})}{{(x^{4} + 27)}^{2}}$

Step 2.2.6

Simplify.

Tap for more steps...

Step 2.2.6.1

Apply the distributive property.

$\frac{4 ((3 x^{4} + 3 \cdot 27) x^{2} - 4 x^{6})}{{(x^{4} + 27)}^{2}}$

Step 2.2.6.2

Apply the distributive property.

$\frac{4 (3 x^{4} x^{2} + 3 \cdot 27 x^{2} - 4 x^{6})}{{(x^{4} + 27)}^{2}}$

Step 2.2.6.3

Apply the distributive property.

$\frac{4 (3 x^{4} x^{2}) + 4 (3 \cdot 27 x^{2}) + 4 (- 4 x^{6})}{{(x^{4} + 27)}^{2}}$

Step 2.2.6.4

Simplify the numerator.

Tap for more steps...

Step 2.2.6.4.1

Simplify each term.

Tap for more steps...

Step 2.2.6.4.1.1

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 2.2.6.4.1.1.1

Move $x^{2}$ .

$\frac{4 (3 (x^{2} x^{4})) + 4 (3 \cdot 27 x^{2}) + 4 (- 4 x^{6})}{{(x^{4} + 27)}^{2}}$

Step 2.2.6.4.1.1.2

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

$\frac{4 (3 x^{2 + 4}) + 4 (3 \cdot 27 x^{2}) + 4 (- 4 x^{6})}{{(x^{4} + 27)}^{2}}$

Step 2.2.6.4.1.1.3

Add $2$ and $4$ .

$\frac{4 (3 x^{6}) + 4 (3 \cdot 27 x^{2}) + 4 (- 4 x^{6})}{{(x^{4} + 27)}^{2}}$

Step 2.2.6.4.1.2

Multiply $3$ by $4$ .

$\frac{12 x^{6} + 4 (3 \cdot 27 x^{2}) + 4 (- 4 x^{6})}{{(x^{4} + 27)}^{2}}$

Step 2.2.6.4.1.3

Multiply $3$ by $27$ .

$\frac{12 x^{6} + 4 (81 x^{2}) + 4 (- 4 x^{6})}{{(x^{4} + 27)}^{2}}$

Step 2.2.6.4.1.4

Multiply $81$ by $4$ .

$\frac{12 x^{6} + 324 x^{2} + 4 (- 4 x^{6})}{{(x^{4} + 27)}^{2}}$

Step 2.2.6.4.1.5

Multiply $- 4$ by $4$ .

$\frac{12 x^{6} + 324 x^{2} - 16 x^{6}}{{(x^{4} + 27)}^{2}}$

Step 2.2.6.4.2

Subtract $16 x^{6}$ from $12 x^{6}$ .

$f'' (x) = \frac{- 4 x^{6} + 324 x^{2}}{{(x^{4} + 27)}^{2}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{- 4 x^{6} + 324 x^{2}}{{(x^{4} + 27)}^{2}}$ .

$\frac{- 4 x^{6} + 324 x^{2}}{{(x^{4} + 27)}^{2}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{- 4 x^{6} + 324 x^{2}}{{(x^{4} + 27)}^{2}} = 0$ .

Tap for more steps...

Step 3.1

Set the second derivative equal to $0$ .

$\frac{- 4 x^{6} + 324 x^{2}}{{(x^{4} + 27)}^{2}} = 0$

Step 3.2

Set the numerator equal to zero.

$- 4 x^{6} + 324 x^{2} = 0$

Step 3.3

Solve the equation for $x$ .

Tap for more steps...

Step 3.3.1

Factor the left side of the equation.

Tap for more steps...

Step 3.3.1.1

Factor $- 4 x^{2}$ out of $- 4 x^{6} + 324 x^{2}$ .

Tap for more steps...

Step 3.3.1.1.1

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

$- 4 x^{2} x^{4} + 324 x^{2} = 0$

Step 3.3.1.1.2

Factor $- 4 x^{2}$ out of $324 x^{2}$ .

$- 4 x^{2} x^{4} - 4 x^{2} \cdot - 81 = 0$

Step 3.3.1.1.3

Factor $- 4 x^{2}$ out of $- 4 x^{2} (x^{4}) - 4 x^{2} (- 81)$ .

$- 4 x^{2} (x^{4} - 81) = 0$

Step 3.3.1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- 4 x^{2} ({(x^{2})}^{2} - 81) = 0$

Step 3.3.1.3

Rewrite $81$ as $9^{2}$ .

$- 4 x^{2} ({(x^{2})}^{2} - 9^{2}) = 0$

Step 3.3.1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 9$ .

$- 4 x^{2} ((x^{2} + 9) (x^{2} - 9)) = 0$

Step 3.3.1.5

Factor.

Tap for more steps...

Step 3.3.1.5.1

Simplify.

Tap for more steps...

Step 3.3.1.5.1.1

Rewrite $9$ as $3^{2}$ .

$- 4 x^{2} ((x^{2} + 9) (x^{2} - 3^{2})) = 0$

Step 3.3.1.5.1.2

Factor.

Tap for more steps...

Step 3.3.1.5.1.2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$- 4 x^{2} ((x^{2} + 9) ((x + 3) (x - 3))) = 0$

Step 3.3.1.5.1.2.2

Remove unnecessary parentheses.

$- 4 x^{2} ((x^{2} + 9) (x + 3) (x - 3)) = 0$

Step 3.3.1.5.2

Remove unnecessary parentheses.

$- 4 x^{2} (x^{2} + 9) (x + 3) (x - 3) = 0$

Step 3.3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$x^{2} + 9 = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 3.3.3

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

Tap for more steps...

Step 3.3.3.1

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

$x^{2} = 0$

Step 3.3.3.2

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

Tap for more steps...

Step 3.3.3.2.1

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

$x = \pm \sqrt{0}$

Step 3.3.3.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 3.3.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.3.3.2.2.2

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

$x = \pm 0$

Step 3.3.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.3.4

Set $x^{2} + 9$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.3.4.1

Set $x^{2} + 9$ equal to $0$ .

$x^{2} + 9 = 0$

Step 3.3.4.2

Solve $x^{2} + 9 = 0$ for $x$ .

Tap for more steps...

Step 3.3.4.2.1

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 3.3.4.2.2

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

$x = \pm \sqrt{- 9}$

Step 3.3.4.2.3

Simplify $\pm \sqrt{- 9}$ .

Tap for more steps...

Step 3.3.4.2.3.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 3.3.4.2.3.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 3.3.4.2.3.3

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

$x = \pm i \cdot \sqrt{9}$

Step 3.3.4.2.3.4

Rewrite $9$ as $3^{2}$ .

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

Step 3.3.4.2.3.5

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

$x = \pm i \cdot 3$

Step 3.3.4.2.3.6

Move $3$ to the left of $i$ .

$x = \pm 3 i$

Step 3.3.4.2.4

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

Tap for more steps...

Step 3.3.4.2.4.1

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

$x = 3 i$

Step 3.3.4.2.4.2

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

$x = - 3 i$

Step 3.3.4.2.4.3

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

$x = 3 i, - 3 i$

Step 3.3.5

Set $x + 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.3.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.3.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.3.6

Set $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.3.6.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.3.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3.3.7

The final solution is all the values that make $- 4 x^{2} (x^{2} + 9) (x + 3) (x - 3) = 0$ true.

$x = 0, 3 i, - 3 i, - 3, 3$

Step 4

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

Tap for more steps...

Step 4.1

Substitute $0$ in $f (x) = \ln (x^{4} + 27)$ to find the value of $y$ .

Tap for more steps...

Step 4.1.1

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

$f (0) = \ln ({(0)}^{4} + 27)$

Step 4.1.2

Simplify the result.

Tap for more steps...

Step 4.1.2.1

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

$f (0) = \ln (0 + 27)$

Step 4.1.2.2

Add $0$ and $27$ .

$f (0) = \ln (27)$

Step 4.1.2.3

The final answer is $\ln (27)$ .

$\ln (27)$

Step 4.2

The point found by substituting $0$ in $f (x) = \ln (x^{4} + 27)$ is $(0, \ln (27))$ . This point can be an inflection point.

$(0, \ln (27))$

Step 4.3

Substitute $- 3$ in $f (x) = \ln (x^{4} + 27)$ to find the value of $y$ .

Tap for more steps...

Step 4.3.1

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

$f (- 3) = \ln ({(- 3)}^{4} + 27)$

Step 4.3.2

Simplify the result.

Tap for more steps...

Step 4.3.2.1

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

$f (- 3) = \ln (81 + 27)$

Step 4.3.2.2

Add $81$ and $27$ .

$f (- 3) = \ln (108)$

Step 4.3.2.3

The final answer is $\ln (108)$ .

$\ln (108)$

Step 4.4

The point found by substituting $- 3$ in $f (x) = \ln (x^{4} + 27)$ is $(- 3, \ln (108))$ . This point can be an inflection point.

$(- 3, \ln (108))$

Step 4.5

Substitute $3$ in $f (x) = \ln (x^{4} + 27)$ to find the value of $y$ .

Tap for more steps...

Step 4.5.1

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

$f (3) = \ln ({(3)}^{4} + 27)$

Step 4.5.2

Simplify the result.

Tap for more steps...

Step 4.5.2.1

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

$f (3) = \ln (81 + 27)$

Step 4.5.2.2

Add $81$ and $27$ .

$f (3) = \ln (108)$

Step 4.5.2.3

The final answer is $\ln (108)$ .

$\ln (108)$

Step 4.6

The point found by substituting $3$ in $f (x) = \ln (x^{4} + 27)$ is $(3, \ln (108))$ . This point can be an inflection point.

$(3, \ln (108))$

Step 4.7

Determine the points that could be inflection points.

$(0, \ln (27)), (- 3, \ln (108)), (3, \ln (108))$

Step 5

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

$(- \infty, - 3) \cup (- 3, 0) \cup (0, 3) \cup (3, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 3)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $- 3.1$ in the expression.

$f'' (- 3.1) = \frac{- 4 {(- 3.1)}^{6} + 324 {(- 3.1)}^{2}}{{({(- 3.1)}^{4} + 27)}^{2}}$

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

Raise $- 3.1$ to the power of $6$ .

$f'' (- 3.1) = \frac{- 4 \cdot 887.503681 + 324 {(- 3.1)}^{2}}{{({(- 3.1)}^{4} + 27)}^{2}}$

Step 6.2.1.2

Multiply $- 4$ by $887.503681$ .

$f'' (- 3.1) = \frac{- 3550.014724 + 324 {(- 3.1)}^{2}}{{({(- 3.1)}^{4} + 27)}^{2}}$

Step 6.2.1.3

Raise $- 3.1$ to the power of $2$ .

$f'' (- 3.1) = \frac{- 3550.014724 + 324 \cdot 9.61}{{({(- 3.1)}^{4} + 27)}^{2}}$

Step 6.2.1.4

Multiply $324$ by $9.61$ .

$f'' (- 3.1) = \frac{- 3550.014724 + 3113.64}{{({(- 3.1)}^{4} + 27)}^{2}}$

Step 6.2.1.5

Add $- 3550.014724$ and $3113.64$ .

$f'' (- 3.1) = \frac{- 436.374724}{{({(- 3.1)}^{4} + 27)}^{2}}$

Step 6.2.2

Simplify the denominator.

Tap for more steps...

Step 6.2.2.1

Raise $- 3.1$ to the power of $4$ .

$f'' (- 3.1) = \frac{- 436.374724}{{(92.3521 + 27)}^{2}}$

Step 6.2.2.2

Add $92.3521$ and $27$ .

$f'' (- 3.1) = \frac{- 436.374724}{{119.3521}^{2}}$

Step 6.2.2.3

Raise $119.3521$ to the power of $2$ .

$f'' (- 3.1) = \frac{- 436.374724}{14244.92377441}$

Step 6.2.3

Divide $- 436.374724$ by $14244.92377441$ .

$f'' (- 3.1) = - 0.0306337$

Step 6.2.4

The final answer is $- 0.0306337$ .

$- 0.0306337$

Step 6.3

At $- 3.1$ , the second derivative is $- 0.0306337$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 3)$

Decreasing on $(- \infty, - 3)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 3, 0)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

$f'' (- \frac{3}{2}) = \frac{- 4 {(- \frac{3}{2})}^{6} + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify the numerator.

Tap for more steps...

Step 7.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 7.2.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$f'' (- \frac{3}{2}) = \frac{- 4 ({(- 1)}^{6} {(\frac{3}{2})}^{6}) + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$f'' (- \frac{3}{2}) = \frac{- 4 ({(- 1)}^{6} (\frac{3^{6}}{2^{6}})) + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.2

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

$f'' (- \frac{3}{2}) = \frac{- 4 (1 (\frac{3^{6}}{2^{6}})) + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.3

Multiply $\frac{3^{6}}{2^{6}}$ by $1$ .

$f'' (- \frac{3}{2}) = \frac{- 4 \frac{3^{6}}{2^{6}} + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.4

Raise $3$ to the power of $6$ .

$f'' (- \frac{3}{2}) = \frac{- 4 \frac{729}{2^{6}} + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.5

Raise $2$ to the power of $6$ .

$f'' (- \frac{3}{2}) = \frac{- 4 (\frac{729}{64}) + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.6

Cancel the common factor of $4$ .

Tap for more steps...

Step 7.2.1.6.1

Factor $4$ out of $- 4$ .

$f'' (- \frac{3}{2}) = \frac{4 (- 1) (\frac{729}{64}) + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.6.2

Factor $4$ out of $64$ .

$f'' (- \frac{3}{2}) = \frac{4 \cdot (- 1 \frac{729}{4 \cdot 16}) + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.6.3

Cancel the common factor.

$f'' (- \frac{3}{2}) = \frac{4 \cdot (- 1 \frac{729}{4 \cdot 16}) + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.6.4

Rewrite the expression.

$f'' (- \frac{3}{2}) = \frac{- 1 (\frac{729}{16}) + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.7

Rewrite $- 1 (\frac{729}{16})$ as $- (\frac{729}{16})$ .

$f'' (- \frac{3}{2}) = \frac{- (\frac{729}{16}) + 324 {(- \frac{3}{2})}^{2}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 7.2.1.8.1

Apply the product rule to $- \frac{3}{2}$ .

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + 324 ({(- 1)}^{2} {(\frac{3}{2})}^{2})}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.8.2

Apply the product rule to $\frac{3}{2}$ .

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + 324 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}}))}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.9

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

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + 324 (1 (\frac{3^{2}}{2^{2}}))}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.10

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + 324 (\frac{3^{2}}{2^{2}})}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.11

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

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + 324 (\frac{9}{2^{2}})}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.12

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

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + 324 (\frac{9}{4})}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.13

Cancel the common factor of $4$ .

Tap for more steps...

Step 7.2.1.13.1

Factor $4$ out of $324$ .

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + 4 (81) (\frac{9}{4})}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.13.2

Cancel the common factor.

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + 4 \cdot (81 (\frac{9}{4}))}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.13.3

Rewrite the expression.

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + 81 \cdot 9}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.14

Multiply $81$ by $9$ .

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + 729}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.15

To write $729$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + 729 \cdot \frac{16}{16}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.16

Combine $729$ and $\frac{16}{16}$ .

$f'' (- \frac{3}{2}) = \frac{- \frac{729}{16} + \frac{729 \cdot 16}{16}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.17

Combine the numerators over the common denominator.

$f'' (- \frac{3}{2}) = \frac{\frac{- 729 + 729 \cdot 16}{16}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.18

Simplify the numerator.

Tap for more steps...

Step 7.2.1.18.1

Multiply $729$ by $16$ .

$f'' (- \frac{3}{2}) = \frac{\frac{- 729 + 11664}{16}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.1.18.2

Add $- 729$ and $11664$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{({(- \frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.2

Simplify the denominator.

Tap for more steps...

Step 7.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 7.2.2.1.1

Apply the product rule to $- \frac{3}{2}$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{({(- 1)}^{4} {(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 7.2.2.1.2

Apply the product rule to $\frac{3}{2}$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{({(- 1)}^{4} (\frac{3^{4}}{2^{4}}) + 27)}^{2}}$

Step 7.2.2.2

Raise $- 1$ to the power of $4$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{(1 (\frac{3^{4}}{2^{4}}) + 27)}^{2}}$

Step 7.2.2.3

Multiply $\frac{3^{4}}{2^{4}}$ by $1$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{3^{4}}{2^{4}} + 27)}^{2}}$

Step 7.2.2.4

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

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81}{2^{4}} + 27)}^{2}}$

Step 7.2.2.5

Raise $2$ to the power of $4$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81}{16} + 27)}^{2}}$

Step 7.2.2.6

To write $27$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81}{16} + 27 \cdot \frac{16}{16})}^{2}}$

Step 7.2.2.7

Combine $27$ and $\frac{16}{16}$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81}{16} + \frac{27 \cdot 16}{16})}^{2}}$

Step 7.2.2.8

Combine the numerators over the common denominator.

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81 + 27 \cdot 16}{16})}^{2}}$

Step 7.2.2.9

Simplify the numerator.

Tap for more steps...

Step 7.2.2.9.1

Multiply $27$ by $16$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81 + 432}{16})}^{2}}$

Step 7.2.2.9.2

Add $81$ and $432$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{513}{16})}^{2}}$

Step 7.2.2.10

Apply the product rule to $\frac{513}{16}$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{\frac{513^{2}}{16^{2}}}$

Step 7.2.2.11

Raise $513$ to the power of $2$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{\frac{263169}{16^{2}}}$

Step 7.2.2.12

Raise $16$ to the power of $2$ .

$f'' (- \frac{3}{2}) = \frac{\frac{10935}{16}}{\frac{263169}{256}}$

Step 7.2.3

Multiply the numerator by the reciprocal of the denominator.

$f'' (- \frac{3}{2}) = \frac{10935}{16} \cdot \frac{256}{263169}$

Step 7.2.4

Cancel the common factor of $729$ .

Tap for more steps...

Step 7.2.4.1

Factor $729$ out of $10935$ .

$f'' (- \frac{3}{2}) = \frac{729 (15)}{16} \cdot \frac{256}{263169}$

Step 7.2.4.2

Factor $729$ out of $263169$ .

$f'' (- \frac{3}{2}) = \frac{729 \cdot 15}{16} \cdot \frac{256}{729 \cdot 361}$

Step 7.2.4.3

Cancel the common factor.

$f'' (- \frac{3}{2}) = \frac{729 \cdot 15}{16} \cdot \frac{256}{729 \cdot 361}$

Step 7.2.4.4

Rewrite the expression.

$f'' (- \frac{3}{2}) = \frac{15}{16} \cdot \frac{256}{361}$

Step 7.2.5

Cancel the common factor of $16$ .

Tap for more steps...

Step 7.2.5.1

Factor $16$ out of $256$ .

$f'' (- \frac{3}{2}) = \frac{15}{16} \cdot \frac{16 (16)}{361}$

Step 7.2.5.2

Cancel the common factor.

$f'' (- \frac{3}{2}) = \frac{15}{16} \cdot \frac{16 \cdot 16}{361}$

Step 7.2.5.3

Rewrite the expression.

$f'' (- \frac{3}{2}) = 15 (\frac{16}{361})$

Step 7.2.6

Combine $15$ and $\frac{16}{361}$ .

$f'' (- \frac{3}{2}) = \frac{15 \cdot 16}{361}$

Step 7.2.7

Multiply $15$ by $16$ .

$f'' (- \frac{3}{2}) = \frac{240}{361}$

Step 7.2.8

The final answer is $\frac{240}{361}$ .

$\frac{240}{361}$

Step 7.3

At $- \frac{3}{2}$ , the second derivative is $0.66481994$ . Since this is positive, the second derivative is increasing on the interval $(- 3, 0)$ .

Increasing on $(- 3, 0)$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(0, 3)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

$f'' (\frac{3}{2}) = \frac{- 4 {(\frac{3}{2})}^{6} + 324 {(\frac{3}{2})}^{2}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Simplify the numerator.

Tap for more steps...

Step 8.2.1.1

Apply the product rule to $\frac{3}{2}$ .

$f'' (\frac{3}{2}) = \frac{- 4 \frac{3^{6}}{2^{6}} + 324 {(\frac{3}{2})}^{2}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.2

Raise $3$ to the power of $6$ .

$f'' (\frac{3}{2}) = \frac{- 4 \frac{729}{2^{6}} + 324 {(\frac{3}{2})}^{2}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.3

Raise $2$ to the power of $6$ .

$f'' (\frac{3}{2}) = \frac{- 4 (\frac{729}{64}) + 324 {(\frac{3}{2})}^{2}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 8.2.1.4.1

Factor $4$ out of $- 4$ .

$f'' (\frac{3}{2}) = \frac{4 (- 1) (\frac{729}{64}) + 324 {(\frac{3}{2})}^{2}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.4.2

Factor $4$ out of $64$ .

$f'' (\frac{3}{2}) = \frac{4 \cdot (- 1 \frac{729}{4 \cdot 16}) + 324 {(\frac{3}{2})}^{2}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.4.3

Cancel the common factor.

$f'' (\frac{3}{2}) = \frac{4 \cdot (- 1 \frac{729}{4 \cdot 16}) + 324 {(\frac{3}{2})}^{2}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.4.4

Rewrite the expression.

$f'' (\frac{3}{2}) = \frac{- 1 (\frac{729}{16}) + 324 {(\frac{3}{2})}^{2}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.5

Rewrite $- 1 (\frac{729}{16})$ as $- (\frac{729}{16})$ .

$f'' (\frac{3}{2}) = \frac{- (\frac{729}{16}) + 324 {(\frac{3}{2})}^{2}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.6

Apply the product rule to $\frac{3}{2}$ .

$f'' (\frac{3}{2}) = \frac{- \frac{729}{16} + 324 (\frac{3^{2}}{2^{2}})}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.7

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

$f'' (\frac{3}{2}) = \frac{- \frac{729}{16} + 324 (\frac{9}{2^{2}})}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.8

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

$f'' (\frac{3}{2}) = \frac{- \frac{729}{16} + 324 (\frac{9}{4})}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.9

Cancel the common factor of $4$ .

Tap for more steps...

Step 8.2.1.9.1

Factor $4$ out of $324$ .

$f'' (\frac{3}{2}) = \frac{- \frac{729}{16} + 4 (81) (\frac{9}{4})}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.9.2

Cancel the common factor.

$f'' (\frac{3}{2}) = \frac{- \frac{729}{16} + 4 \cdot (81 (\frac{9}{4}))}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.9.3

Rewrite the expression.

$f'' (\frac{3}{2}) = \frac{- \frac{729}{16} + 81 \cdot 9}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.10

Multiply $81$ by $9$ .

$f'' (\frac{3}{2}) = \frac{- \frac{729}{16} + 729}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.11

To write $729$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$f'' (\frac{3}{2}) = \frac{- \frac{729}{16} + 729 \cdot \frac{16}{16}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.12

Combine $729$ and $\frac{16}{16}$ .

$f'' (\frac{3}{2}) = \frac{- \frac{729}{16} + \frac{729 \cdot 16}{16}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.13

Combine the numerators over the common denominator.

$f'' (\frac{3}{2}) = \frac{\frac{- 729 + 729 \cdot 16}{16}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.14

Simplify the numerator.

Tap for more steps...

Step 8.2.1.14.1

Multiply $729$ by $16$ .

$f'' (\frac{3}{2}) = \frac{\frac{- 729 + 11664}{16}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.1.14.2

Add $- 729$ and $11664$ .

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{{({(\frac{3}{2})}^{4} + 27)}^{2}}$

Step 8.2.2

Simplify the denominator.

Tap for more steps...

Step 8.2.2.1

Apply the product rule to $\frac{3}{2}$ .

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{3^{4}}{2^{4}} + 27)}^{2}}$

Step 8.2.2.2

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

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81}{2^{4}} + 27)}^{2}}$

Step 8.2.2.3

Raise $2$ to the power of $4$ .

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81}{16} + 27)}^{2}}$

Step 8.2.2.4

To write $27$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81}{16} + 27 \cdot \frac{16}{16})}^{2}}$

Step 8.2.2.5

Combine $27$ and $\frac{16}{16}$ .

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81}{16} + \frac{27 \cdot 16}{16})}^{2}}$

Step 8.2.2.6

Combine the numerators over the common denominator.

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81 + 27 \cdot 16}{16})}^{2}}$

Step 8.2.2.7

Simplify the numerator.

Tap for more steps...

Step 8.2.2.7.1

Multiply $27$ by $16$ .

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{81 + 432}{16})}^{2}}$

Step 8.2.2.7.2

Add $81$ and $432$ .

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{{(\frac{513}{16})}^{2}}$

Step 8.2.2.8

Apply the product rule to $\frac{513}{16}$ .

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{\frac{513^{2}}{16^{2}}}$

Step 8.2.2.9

Raise $513$ to the power of $2$ .

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{\frac{263169}{16^{2}}}$

Step 8.2.2.10

Raise $16$ to the power of $2$ .

$f'' (\frac{3}{2}) = \frac{\frac{10935}{16}}{\frac{263169}{256}}$

Step 8.2.3

Multiply the numerator by the reciprocal of the denominator.

$f'' (\frac{3}{2}) = \frac{10935}{16} \cdot \frac{256}{263169}$

Step 8.2.4

Cancel the common factor of $729$ .

Tap for more steps...

Step 8.2.4.1

Factor $729$ out of $10935$ .

$f'' (\frac{3}{2}) = \frac{729 (15)}{16} \cdot \frac{256}{263169}$

Step 8.2.4.2

Factor $729$ out of $263169$ .

$f'' (\frac{3}{2}) = \frac{729 \cdot 15}{16} \cdot \frac{256}{729 \cdot 361}$

Step 8.2.4.3

Cancel the common factor.

$f'' (\frac{3}{2}) = \frac{729 \cdot 15}{16} \cdot \frac{256}{729 \cdot 361}$

Step 8.2.4.4

Rewrite the expression.

$f'' (\frac{3}{2}) = \frac{15}{16} \cdot \frac{256}{361}$

Step 8.2.5

Cancel the common factor of $16$ .

Tap for more steps...

Step 8.2.5.1

Factor $16$ out of $256$ .

$f'' (\frac{3}{2}) = \frac{15}{16} \cdot \frac{16 (16)}{361}$

Step 8.2.5.2

Cancel the common factor.

$f'' (\frac{3}{2}) = \frac{15}{16} \cdot \frac{16 \cdot 16}{361}$

Step 8.2.5.3

Rewrite the expression.

$f'' (\frac{3}{2}) = 15 (\frac{16}{361})$

Step 8.2.6

Combine $15$ and $\frac{16}{361}$ .

$f'' (\frac{3}{2}) = \frac{15 \cdot 16}{361}$

Step 8.2.7

Multiply $15$ by $16$ .

$f'' (\frac{3}{2}) = \frac{240}{361}$

Step 8.2.8

The final answer is $\frac{240}{361}$ .

$\frac{240}{361}$

Step 8.3

At $\frac{3}{2}$ , the second derivative is $0.66481994$ . Since this is positive, the second derivative is increasing on the interval $(0, 3)$ .

Increasing on $(0, 3)$ since $f'' (x) > 0$

Step 9

Substitute a value from the interval $(3, \infty)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 9.1

Replace the variable $x$ with $3.1$ in the expression.

$f'' (3.1) = \frac{- 4 {(3.1)}^{6} + 324 {(3.1)}^{2}}{{({(3.1)}^{4} + 27)}^{2}}$

Step 9.2

Simplify the result.

Tap for more steps...

Step 9.2.1

Simplify the numerator.

Tap for more steps...

Step 9.2.1.1

Raise $3.1$ to the power of $6$ .

$f'' (3.1) = \frac{- 4 \cdot 887.503681 + 324 \cdot {3.1}^{2}}{{({3.1}^{4} + 27)}^{2}}$

Step 9.2.1.2

Multiply $- 4$ by $887.503681$ .

$f'' (3.1) = \frac{- 3550.014724 + 324 \cdot {3.1}^{2}}{{({3.1}^{4} + 27)}^{2}}$

Step 9.2.1.3

Raise $3.1$ to the power of $2$ .

$f'' (3.1) = \frac{- 3550.014724 + 324 \cdot 9.61}{{({3.1}^{4} + 27)}^{2}}$

Step 9.2.1.4

Multiply $324$ by $9.61$ .

$f'' (3.1) = \frac{- 3550.014724 + 3113.64}{{({3.1}^{4} + 27)}^{2}}$

Step 9.2.1.5

Add $- 3550.014724$ and $3113.64$ .

$f'' (3.1) = \frac{- 436.374724}{{({3.1}^{4} + 27)}^{2}}$

Step 9.2.2

Simplify the denominator.

Tap for more steps...

Step 9.2.2.1

Raise $3.1$ to the power of $4$ .

$f'' (3.1) = \frac{- 436.374724}{{(92.3521 + 27)}^{2}}$

Step 9.2.2.2

Add $92.3521$ and $27$ .

$f'' (3.1) = \frac{- 436.374724}{{119.3521}^{2}}$

Step 9.2.2.3

Raise $119.3521$ to the power of $2$ .

$f'' (3.1) = \frac{- 436.374724}{14244.92377441}$

Step 9.2.3

Divide $- 436.374724$ by $14244.92377441$ .

$f'' (3.1) = - 0.0306337$

Step 9.2.4

The final answer is $- 0.0306337$ .

$- 0.0306337$

Step 9.3

At $3.1$ , the second derivative is $- 0.0306337$ . Since this is negative, the second derivative is decreasing on the interval $(3, \infty)$

Decreasing on $(3, \infty)$ 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 $(- 3, \ln (108)), (3, \ln (108))$ .

$(- 3, \ln (108)), (3, \ln (108))$

Step 11