Rolle's theorem
Teorema 6.0 (Rolle's theorem) - Let f: [a, b] → ℝ, continuous on the interval [a,b] and at least differentiable on (a,b). If f(a) = f(b), then there exist at least a point x0 ∈ (a,b) such that:
f'(x0) = 0
Proof By Extreme Value Theorem, if a function f is continuous on a closed interval then attains absolute maximum and aablosute minimum. There are two cases to consider:
f is constant, thus: m = M = f(a) = f(b), and f'(x) = 0, ∀x ∈ (a,b).
If m < M, at least one of the two extremes is internal to [a,b], and by Fermat's stationary point theorem f'(x) = 0.
Applicability of Rolle's theorem
Rolle's theorem holds only when all the three conditions are satisfied on the interval under consideration. Even if one of the conditions fails to hold, then the conclusion of the Rolle's Theorem is not applicable. Consider the absolute value function:
f(x) = |x|, x ∈ [−1,1]
it results that f(1) = f(−1), but the function is not differentiable in x = 0, thus the hypothesis of Rolle's theorem do not hold.
