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Chapter 3: Q. 54 (page 249)

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)=x3−2x+1,[a,b]=[0,6]

Short Answer

Expert verified

The function satisfies the mean value theorem andc=±23

Step by step solution

01

Step 1. Given information

f(x)=x3−2x+1

02

Step 2. Proving Mean Value Theorem 

f′(c)=f(b)−f(a)b−a=63−2⋅6+1−03−2⋅0+16−0=216−12+1−16=34

Now,

role="math" localid="1648384027018" f′(c)=343c2−2=343c2=36c2=12c=±12c=±23

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Most popular questions from this chapter

Find the possibility graph of its derivative f'.

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)=(x−2)(x+2)

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

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

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.
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