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: If f is continuous on [a,b], f is differentiable on (a,b), and f(a) = f(b), then there is c in (a,b) such that f ’(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