Q. 33 - Calculus

Question

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{l}\mathit{n}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}$

Step-by-Step Solution

Answer

Ans: 

After inserting the root values we can find the increasing and decreasing intervals of the given function. 

Intervals of the given function are 

Increasing at $\left(0,\infty \right)$

Decreasing at$\left(-\infty ,0\right)$

Step 1. Given information:

$\mathit{l}\mathit{n}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}$

Step 2. Finding the derivative of the function:

$$\begin{gathered} f\left(x\right)=\ln \left(x^2+1\right) \\ {f}^{\text{'}}\left(x\right)=\frac{2x}{x^2+1} \\ let f^{\text{'}}\left(x\right)=0 \\ \therefore \frac{2x}{x^2+1}=0 \\ \Rightarrow 2x=\left(x^2+1\right)0 \\ \Rightarrow 2x=0 \\ x=0 \end{gathered}$$

Step 3. Inserting the root points on the number line(Sign chart):

After inserting the root values we can find the increasing and decreasing intervals of the given function. 

Intervals of the given function are 

Increasing at $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Decreasing at$\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{0}\mathbf{)}$

Step 4. Verifying algebraic answers with graphs :