Question

How do you determine whether Rolle’s Theorem can be applied to `f(x)=x(x-6)^2` on the interval [0,6]?

Answer 1

See the explanation section.

Explanation:

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?"

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

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

`f(x) = x(x-6)^2` on the interval `[0,6]`

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 `[0,6]` [Because, this function is a polynomial so it is continuous at every real number.]

H2 : The function `f` in this problem is differentiable on `(0,6)`
[Because the derivative, `f'(x) = (x-6)^2+2x(x-6)` exists for all real `x`. In particular, it exists for all `x` in `(0,6)`.)

H3 : `f(0) = 0 = f(6)`

Therefore we can apply Rolle's Theorem to `f(x) = x(x-6)^2` on the interval `[0,6]`. (Meaning "the hypotheses are true.)

Extra
Because the hypotheses are true, we know without further work, that the conclusion of Rolle's Theorem. That is, we know that there is a `c` (at least one `c`) in `(0,6)` where `f'(c) = 0`.

Sometimes, as an equation solving exercise, students are asked to find the `c` that works on the conclusion. This means setting `f'(x)` equal to `0` and selecting the solution or solutions in the open interval.
In this question, `c=2`. Although `f'(6)=0`, `6` is not in the open interval `(0,6)`, so it is not a `c` mentioned in the conclusion.)