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?

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:

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

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,

To determine if Rolle's Theorem can be applied to (f(x) = x^3 - x^2 - 20x + 7) on the interval ([0, 5]), first, check if the function satisfies the conditions of Rolle's Theorem:

1. (f(x)) must be continuous on the closed interval ([0, 5]).
2. (f(x)) must be differentiable on the open interval ((0, 5)).
3. (f(0) = f(5)).

If all three conditions are met, then Rolle's Theorem can be applied.

For the second part of the question, to find all values of (c) in the interval ([0, 5]) for which (f'(c) = 0), follow these steps:

1. Calculate the derivative of (f(x)), (f'(x)).
2. Set (f'(x) = 0) and solve for (x) to find critical points.
3. Check which critical points lie in the interval ([0, 5]).
4. The values of (c) for which (f'(c) = 0) within the interval ([0, 5]) are the critical points found in step 3.