# How do you determine if rolles theorem can be applied to #f(x) = x^3 - x^2- 20x + 7 # on the interval [0,5] and if so how do you find all the values of c in the interval for which f'(c)=0?

##### 1 Answer

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

Rolle's Theorem has three hypotheses:

**H1** :

**H2** :

**H3** :

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

Is the function in this question continuous on the interval

Is it differentiable on the open interval

Is

If the answer to all three is "yes", then the hypotheses are true and we say that Rolle's Theroem "can be applied".

To find all the values of c in the interval for which f'(c)=0,

Find