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Chapter 3: Q. 34 (page 261)

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)=ln(lnx)

Short Answer

Expert verified

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

Intervals of the given function are

Increasing at(1,∞)

Step by step solution

01

Step 1. Given information:

f(x)=ln(ln(x))

02

Step 2. Finding the derivative of the function:

f(x)=lnlnxf'(x)=1xlnxletf'(x)=0∴1xlnx=0⇒10=xlnx⇒xlnx=∞x=∞(nosolutions)

03

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(1,∞)

04

Step 4. Verifying algebraic answers with graphs :

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Most popular questions from this chapter

Determine whether or not each function f in Exercises 53–60 satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of the Mean Value Theorem.

f(x)=x2+1x,[a,b]=[−3,2]

Use the definitions of increasing and decreasing to argue that f(x)=x4 is decreasing on (−∞,0] and increasing on [0,∞). Then use derivatives to argue the same thing.

Last night at 6 p.m., Linda got up from her blue easy chair. She did not return to her easy chair until she sat down again at 8 p.m. Let s(t) be the distance between Linda and her easy chair t minutes after 6 p.m. last night.

(a) Sketch a possible graph of s(t), and describe what Linda did between 6 p.m. and 8 p.m. according to your graph. (Questions to think about: Will Linda necessarily move in a continuous and differentiable way? What are good ranges for t and s?

(b) Use Rolle’s Theorem to show that at some point between 6 p.m. and 8 p.m., Linda’s velocity v(t) with respect to the easy chair was zero. Find such a place on the graph of s(t).

Prove that the function is increasing on all values of real numbers.

f(x)=x3.

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f,f',andf'', and examine any relevant limits so that you can describe all key points and behaviors of f.

f(x)=1-x4-2
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