Q. 8 - Calculus

Question

Sign analyses for second derivatives: Repeat the instructions of the previous block of problems, except find sign intervals for the second derivative $f\text{'}\text{'}$ instead of the first derivative.

$f\left(x\right)=\ln \left(\ln x\right)$

Step-by-Step Solution

Answer

The second derivative of the function has no inflection points.

Step 1. Given Information.

The given function is $f\left(x\right)=\ln \left(\ln x\right).$

Step 2. Find the second derivative.

To find the second derivative, first, we find the first derivative then the second.

So,

$$\begin{gathered} f\left(x\right)=\ln \left(\ln x\right) \\ f\text{'}\left(x\right)=\frac{1}{x\left(\ln x\right)} \\ f\text{'}\text{'}\left(x\right)=\frac{-\left[x\cdot \frac{1}{x}+\ln x\right]}{x^2(\ln x{)}^2} \\ f\text{'}\text{'}\left(x\right)=\frac{-|1+\ln x\mid }{x^2(\ln x{)}^2} \end{gathered}$$

Step 3. Find sign intervals for the second derivative.

To find the sign intervals let's find the critical points from the first derivative.

So,

$$\begin{gathered} f\text{'}\left(x\right)=\frac{1}{x\left(\ln x\right)} \\ 0=\frac{1}{x\left(\ln x\right)} \\ 0\ne 1 \end{gathered}$$

Thus, there aren't any critical points.