How do you determine if rolles theorem can be applied to #f(x) = 2x^3 - x^2 - 8x + 4# on the interval [-2,2] and if so how do you find all the values of c in the interval for which f'(c)=0?

1 Answer
May 25, 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) = 2x^3-x^2-8x+4# , #a=-2# and #b=2#.

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

So answer each question:

H1 : Is #f(x) = 2x^3-x^2-8x+4# continuous on the closed interval #[-2,2]#?

H2 : Is #f(x) = 2x^3-x^2-8x+4# differentiable on the openinterval #(-2,2)#?

H3 : #Is f(-2)=f(2)#?

If the answer to all three questions is yes, then Rolle's can be applied to this function on this interval.

To solve #f'(c) = 0#, find #f'(x)#, set it equal to #0# and solve the equation. (There may be more than one solution.)
Select, as #c#, any solutions in #(-2,2)#
(That is where Rolle's says there must be a solution. There may be more than one solution in the interval.)