Question

How do you determine if Rolles Theorem applies to the given function `x^3 - 9x` on [0,3]. If so, how do you find all numbers c on the interval that satisfy the theorem?

Answer 1

Please see below.

Explanation:

When we are asked whether some theorem "applies" 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^3-9x` on the interval `[0,3]`

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 `[0,3]` [Because, this function is a polynomial so it is continuous at every real number.]

H2 : The function `f` in this problem is differentiable on `(0,3)`
[Because the derivative, `f'(x) =3x^2-9` exists for all real `x`. In particular, it exists for all `x` in `(0,3)`.)

H3 : `f(0) = 0 = f(3)`

Therefore Rolle's Theorem does apply to `f(x) = x^3-9x` on the interval `[0,3]`. (Meaning "the hypotheses are true".)

Extra
Because the hypotheses are true, we know without further work, that the conclusion of Rolle's Theorem must also be true. That is, we know that there is a `c` (at least one `c`) in `(0,3)` where `f'(c) = 0`.

To try to find the value(s) of `c` the theorem tells us are there, we try to solve `f'(x) = 0` on the interval `(0,3)`. (We can be certain that there is a solution, but we may have have trouble finding/expressing the solutions.)

`3x^2-9 = 0` if and only if `x = +- sqrt3`

`-sqrt3` is not in the interval, so the only value of `c` that satisfies the conclusion of Rolle's Theorem is `c=sqrt3`