Question

Does rolle's theorem apply for `f(x) = abs(x-2)` on [0,4]?

Answer 1

No.

Although `f` is continuous on `[0,4]` and `f(0)=f(4)`, we cannot apply Rolle's Theorem because `f` is not differentiable at `2`.

An absolute value function is not differentiable at its vertex. In this case `f'(x) = -1` left of `2` and `f'(x)=1` right of `2`. `f'(2)` does not exist.