Question

How do you verify whether rolle's theorem can be applied to the function `f(x)=x^3` in [1,3]?

Answer 1

When we are asked whether some theorem "can be applied" to some situation, we are really being asked "Are the hypotheses of the theorem true for this situation?"

Explanation:

(The hypotheses are also called the antecedent, of 'the if parts'.)

So we need to determine whether the hypotheses of Rolle's Theorem are true for the function

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

Rolle's Theorem has three hypotheses:

H1 : `f` is continuous on the closed interval `[a,b]`
H2 : `f` is differentiable on the open interval `(a,b)`.
H3 : `f(a)=f(b)`

We say that we can apply Rolle's Theorem if all 3 hypotheses are true.

H1 : The function `f` in this problem is continuous on `[1,3]` [Because, this function is a polynomial so it is continuous at every real number.]

H2 : The function `f` in this problem is differentiable on `(1,3)`
[Because the derivative, `f'(x) = 3x^2` exists for all real `x`. In particular, it exists for all `x` in `(1,3)`.)

H3 : `f(1) = 1^3 = 1` and `f(3) = 3^3 = 27`.
No, we do not have `f(a)=f(b)`. The third hypothesis is false for this function on this interval.

Therefore we cannot apply Rolle's Theorem to `f(x) = x^3` on the interval `[1,3]`. (Meaning "at least one hypothesis is false".).)