# How do you verify that the function #f(x)=x^3 - 21x^2 + 80x + 2# satisfies Rolle's Theorem on the given interval [0,16] and then find all numbers c that satisfy the conclusion of Rolle's Theorem?

##### 1 Answer

Rolle's Theorem has three hypotheses:

**H1** :

**H2** :

**H3** :

In this question,

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

So answer each question:

**H1** : Is

**H2** : Is

**H3** : Is

If the answer to all three questions is yes, then Rolle's can be applied to this function on this interval.

To solve

Select, as

(That is where Rolle's says there must be a solution. There may be more than one solution in the interval.)

**Arithmetic note:**

since

so

Please don't waste any of your youth finding