Q. 34 - 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.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{l}\mathit{n}\left(ln\left(x\right)\right)$

Step-by-Step Solution

Answer

Ans: It has no solution hence this is a constant function.

Intervals of the given function are 

Increasing at $(1,\infty )$

Step 1. Given information:

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{l}\mathit{n}\mathbf{\left(}\mathbf{ln}\left(x\right)\mathbf{\right)}$

Step 2. Finding the derivative of the function:

$$\begin{gathered} f\left(x\right)=\ln \left(\ln \left(x\right)\right) \\ {f}^{\text{'}}\left(x\right)=\frac{1}{x\ln \left(x\right)} \\ let f^{\text{'}}\left(x\right)=0 \\ \therefore \frac{1}{x\ln \left(x\right)}=0 \\ \Rightarrow \frac{1}{0}=x\ln \left(x\right) \\ \Rightarrow x\ln \left(x\right)=\infty \\ x=\infty (no solutions) \end{gathered}$$

Step 3. Finding increasing and decreasing intervals:

It has no solution hence this is a constant function.

so according to the graph, we can say that the function increases from 1 

Therefore$\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Step 4. Verifying algebraic answers with graphs :