Question

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?

Answer 1

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^3 - x^2- 20x + 7` on the interval `[0,5]`

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.

Is the function in this question continuous on the interval `[0,5]`?

Is it differentiable on the open interval `(0,5)`?

Is `f(0)=f(5)`.

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 `f'(x)`, set it equal to `0`, solve the equation list the solutions that are in the interval `(0,5)`.