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?

1 Answer
May 1, 2015

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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