CALCULUS Understanding Its Concepts and Methods

Rolle's theorem

Rolle's
theoremLet
*f*
be a function such that

1. |
f
is continuous on the closed interval
[a,b] |

2. |
f
is differentiable on the open interval
(a,b) |

3. |
f(a) = f(b) |

Then there is a number
*c*
in
(*a*,*b*)
for which
*f'*(*c*) = 0.

Historically, condition **3** was
*f*(*a*) = *f*(*b*) = 0, but
the restriction is superfluous.

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.