Question

How do you determine whether rolles theorem can be applied to `f(X)=(x^2-2x-3)/(x+2)` on the closed interval [-1,3] and if it can find all value of c such that f '(c)=0?

Answer 1

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)`

In this question, `f(x) = (x^2-2x-3)/(x+2)` , `a=-1` and `b=3`.

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

H1 : Is this function is continuous on `[-1,3]`?

H2 : Is the derivative of this function defined on `(-1,3)`?

If so, then `f` is differentiable on `(-1, 3)`

H3 : Is `f(-1) = f(3)`?

If the answer to all 3 questions is "yes", then we can apply Rolle's Theorem to this function on this interval.
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As an algebra exercise you've also been asked to actually find the values of `c` whose existence is guaranteed by Rolle's Theorem.

To do that, find `f'(x)`, set it equal to `0` and do the algebra to solve the equation.
Select any solutions in `(-1, 3)` as values of `c`.
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