Question

How do you know why Rolle's Theorem does not apply to `f(x)= x^(2/3)` on the interval [-1,1]?

Answer 1

`f(x)= x^(2/3)` on the interval [-1,1]

H1
Is the function continuous on the closed interval?

Yes, power functions are continuous on their domains.

H2
Is the function differentiable on the open interval?

No.

`f'(x) = 2/(3 root(3) x)` does not exist at `x = 0`. The function is not differentiable on any interval that includes `0`

(By the way, H3 `f(-1)=f(1)` is true for this function.)

Also by the way: This function does not satisfy the conclusion of Rolle's Theorem on any interval, because there is no solution to:

`f'(x) = 2/(3 root(3) x) = 0`