Rolle's theorem

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Statement

Suppose f is a function defined on a closed interval [a,b] (with a < b) satisfying the following three conditions:

  1. f is a continuous function on the closed interval [a,b]. In particular, f is (two-sided) continuous at every point in the open interval (a,b), right continuous at a, and left continuous at b.
  2. f is differentiable on the open interval (a,b), i.e., the derivative of f exists at all points in the open interval (a,b).
  3. f(a) = f(b) = 0.

Then, there exists c in the open interval (a,b) such that f'(c) = 0.

Related facts

Applications

Facts used

  1. Extreme value theorem
  2. Point of local extremum implies critical point

Proof

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 If f is zero on all of [a,b], then f'(c) = 0 for any choice of c\in (a,b) obvious
2 f must attain both its maximum and its minimum values on [a,b] Fact (1) f is continuous on [a,b] Given-fact combination direct
3 If f is not zero on all of [a,b], either its absolute maximum value on [a,b] is positive and attained at a point in the open interval (a,b) or its absolute minimum value on [a,b] is negative and attained at a point in the open interval (a,b) (or possibly both). f(a) = f(b) = 0 Step (2) [SHOW MORE]
4 If c is a point in (a,b) at which f attains its maximum value or its minimum value, then \! f'(c) = 0. Fact (2) f is differentiable on (a,b) [SHOW MORE]
5 There is a point c\in (a,b) at which \! f'(c) = 0. Steps (1), (3), (4) [SHOW MORE]