L2RolleMean

# L2RolleMean - ROLLES THEOREM AND THE MEAN VALUE THEOREM...

This preview shows pages 1–2. Sign up to view the full content.

ROLLE’S THEOREM AND THE MEAN VALUE THEOREM WILLIAM A. LAMPE Recall the Theorem on Local Extrema. If f ( c ) is a local extremum, then either f is not differentiable at c or f ( c ) = 0 . That is, at a local max or min f either has no tangent, or f has a horizontal tangent there. We will use this to prove Rolle’s Theorem. Let a < b . If f is continuous on the closed interval [ a,b ] and differen- tiable on the open interval ( a,b ) and f ( a ) = f ( b ) , then there is a c in ( a,b ) with f ( c ) = 0 . That is, under these hypotheses, f has a horizontal tangent somewhere between a and b . Rolle’s Theorem, like the Theorem on Local Extrema, ends with f ( c ) = 0. The proof of Rolle’s Theorem is a matter of examining cases and applying the Theorem on Local Extrema. Proof. We seek a c in ( a,b ) with f ( c ) = 0. That is, we wish to show that f has a horizontal tangent somewhere between a and b . Since f is continuous on the closed interval [ a,b ], the Extreme Value Theorem says that f has a maximum value f ( M ) and a minimum value f ( m ) on the closed interval [ a,b ]. Either f ( M ) = f ( m ) or f ( M ) negationslash = f ( m ). Case 1. We suppose the maximum value f ( M ) = f ( m ), the minimum value. So all values of f on [ a,b ] are equal, and f is constant on [ a,b ]. Then f ( x ) = 0 for all x in ( a,b ). So one may take c to be anything in ( a,b ); for example, c = a + b 2 would suffice. Case 2. Now we suppose f ( M ) negationslash = f ( m ). So at least one of f ( M ) and f ( m ) is not equal to the value f ( a ) = f ( b ). Case 2.a We first consider the case where the maximum value f ( M ) negationslash = f ( a ) = f ( b ). (See the figure to the right.) So M is neither a nor b . But M is in [ a,b ] and not at the end points. So M must be in the open interval ( a,b ). We have the maximum value f ( M ) f ( x ) for all x in the closed interval[ a,b

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern