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Chapter 3: Q. 65 (page 249)

Prove Rolle鈥檚 Theorem: If ff is continuous on [a,b]and differentiable on (a,b), and if f(a)=f(b)=0, then there is some value c鈭�(a,b)with f(c)=0.

Short Answer

Expert verified

We have proved the statement is true for Rolle's Theorem.

Step by step solution

01

Step 1. Given Information.

The function fis given.

02

Step 2. Rolle's Theorem.

If fis continuous on the closed interval [a,b]and differentiable on the open interval (a,b)with f(a)=f(b)=0. We know that fattains both a maximum and a minimum value on [a,b]. If one of these extreme values occur at x=cin the interior (a,b)of the interval, then x=cis a local extremum of f. Using the theorem of "Local Extrema are Critical Points" this means that x=cis a critical point of f. Hence, it follows that f'(c)=0as fis assumed to be differentiable at x=c. Hence, Rolle's Theorem is verified.

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Most popular questions from this chapter

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)=cos2(x)

Sketch careful, labeled graphs of each function f in Exercises 63鈥�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)=x2-xx2-3x+2

Sketch careful, labeled graphs of each function f in Exercises 63鈥�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)=x2-1x2-5x+4

Sketch careful, labeled graphs of each function f in Exercises 63鈥�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)=lnx2+1

Find the possibility graph of its derivative f'.
See all solutions

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