Question

How do you find the number c that satisfies the conclusion of Rolle's Theorem `f(x) = x^3 - x^2 - 20x + 3` on interval [0, 5]?

Answer 1

Solve `f'(x) = 0`. At least one solution will be in `(0,5)`

Explanation:

It is best to first check: this `f` is a polynomial function, so it is continuous and differentiable everywhere. Furhermore `f(0) = f(5) = 3`.
This means that this function on this intevral satisfies the hypotheses for Rolle's Theorem.
So the conclusion of Rolle's Theorem must also be true in this case.

The conclusion of Rolle's Theorem says there is a `c` in `(0,5)` with `f'(c) =0`. We have been asked to find the values of `c` that this conclusion refers to.

`f(x) = x^3 - x^2 - 20x + 3`

`f'(x) = 3x^2 - 2x - 20`

`3x^2 - 2x - 20 = 0`

at `x = (1 +- sqrt61)/3`

We know that `7 < sqrt61 < 8`,

so `(1 - sqrt61)/3` is negative. (It is not in `(0,5)`)

We can trust the theorem and conclude that the `c` we are looking for is `(1 + sqrt61)/3` or, if we want independent verification:

`7 < sqrt61 < 8`, so `(1+7)/3 < (1 + sqrt61)/3 < (1+8)/3`

therefore `(1 + sqrt61)/3` is between `8/3 = 2 2/3` and `9/3 = 3` and it is in `(0,5)`.