Question

How do you know whether Rolle's Theorem applies for `f(x)=x^2/3+1` on the interval [-1,1]?

Answer 1

Ask your self whether the hypotheses of Rolle's Theorem are true for this function on this interval:

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

H1

Is `f` continuous on `[-1,1]`?

Yes, `f` is a polynomial (quadratic) hence, continuous everywhere. So, in particular, `f` is continuous on `[-1,1]`

H2

Is `f` differentiable on `(-1,1)`?

Yes, `f'(x) = (2x)/3` exists for all `x` in `(-1,1)`
(In fact, `f'(x)` exists for all real `x`, but the theorem only requires the interval.)

H3

Is `f(-1) = f(1)`?

Yes, either by arithimetic (`f(-1) = 4/3 = f(1)`

or by using the fact that `f` is an even function, so f(-x) = f(x) for every `x`.

Because it satisfies all of the hypotheses, we say that the Theorem applies to this function on this interval.