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Chapter 3: Q. 8 (page 250)

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

f(x)=lnlnx

Short Answer

Expert verified

The second derivative of the function has no inflection points.

Step by step solution

01

Step 1. Given Information.

The given function isf(x)=lnlnx.

02

Step 2. Find the second derivative. 

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

So,

f(x)=ln(lnx)f'(x)=1x(lnx)f''(x)=鈭�x鈰�1x+lnxx2(lnx)2f''(x)=鈭�|1+lnx鈭�x2(lnx)2

03

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,

f'(x)=1x(lnx)0=1x(lnx)0鈮�1

Thus, there aren't any critical points.

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Most popular questions from this chapter

The cost of manufacturing a container for frozen orange juice is C(h)=h2-7.4h+13.7cents, where his the height of the container in inches. Your boss claims that the containers will be the cheapest to make if they are 4inches tall. Use Theorem 3.3 to quickly show that he is wrong.

Use a sign chart for f'to determine the intervals on which each function fis increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

f(x)=1sinx

Find the possibility graph of its derivative f'.

Determine the graph of a function f from the graph of its derivative f'.

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.
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