Solution - Examine the Applicability of Mean Value Theorem for All Three Functions Given in the Above Exercise 2. - Mean Value Theorem

Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?

f (x) = [x] for x ∈ [5, 9]

If f : [– 5, 5] → R is a differentiable function and if f ′(x) does not vanish anywhere, then prove that f (– 5) ≠ f (5).

Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?

f (x) = x² – 1 for x ∈ [1, 2]