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Chapter 3: Q. 55 (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+3x2−7,[a,b]=[−2,3]

Short Answer

Expert verified

The function satisfies the mean value theorem andc=2.527,−0.527

Step by step solution

01

Step 1. Given information

f(x)=−x3+3x2−7

02

Step 2. Proving Mean Value Theorem 

f′(c)=f(3)−f(−2)3−(−2)=−33+3⋅32−7−−(−2)3+3⋅(−2)2−75=−7−8−12+75=−4

Now,

f′(c)=−4−3c2+3⋅2c=−43c2−6c−4=0

c=−(−6)±(−6)2−4⋅3⋅(−4)2⋅3=6±846

=6±9.1656=2.527,−0.527

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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=3x-2ex

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)=1x2+3

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)=exx

Restate Theorem 3.3 so that its conclusion has to do with

tangent lines.

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

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