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Chapter 3: Q. 47 (page 261)

Use the first-derivative test to determine the local extrema of each function fin Exercises 39-50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

fx=arctanx.

Short Answer

Expert verified

The function fx=arctanxhas no local extrema. The following graph verifies the algebraic result graphically:

Step by step solution

01

Step 1. Given information

fx=arctanx.

02

Step 2. Consider the function,

fx=arctanx.

First find the derivative for the given function.

f'x=ddxarctanx=11+x2

The derivative is defined and continuous everywhere, so the critical points of fare just the points where role="math" localid="1648397112503" f'x=0that is,

f'x=0

Therefore, there is no critical point for which f'(x)=0.

Hence, there is no local extrema.

03

Step 3. The following graph verifies the algebraic result graphically:

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Most popular questions from this chapter

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

fx=3x4+8x3-18x2

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-xex

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

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).

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

f(x)=(x-1)2x+2
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