# Rolle's Theorem

### Equal endpoints + differentiable = derivative is zero somewhere in between.

Hi Scholars!

This week we review a result many of us briefly saw in calculus:

Rolle’s Theorem: Iffis continuous on [a,b],fis differentiable on (a,b), andf(a) =f(b), then there iscin (a,b) such thatf’(c) = 0.

Upon drawing a picture, we can often intuit that this result holds. And, it is perhaps one of the more intuitive results to verify formally. If the maximum and minimum of *f* are at the endpoints, then *f *is constant and the result follows. Otherwise, by the extreme value theorem, we know there is a point somewhere in (*a*,*b*) where *f *attains either a maximum or a minimum. In the picture, we assume there is a maximum at a point *c*. At any point *x *< *c, *the slope of a secant line from (*x*, *f*(*x*)) to (*c*, *f*(*c*)) is nonnegative. Thus, letting x approach c reveals the left hand limit is nonnegative too. But, this limit is precisely the derivative, and so *f* ‘(*c* ) ≥ 0. Similar argument applies with the right hand limit to deduce *f'*(c) ≤ 0. Combining these inequalities, we conclude *f* ‘(*c*) = 0, as desired. That outlines all the key argument steps!

Stay Awesome.

Howard