Continuity and Differentiability
Okay can somebody please check my answer to this
The function f is continuous for [-2,1] and differentiable for (-2,1). If f(-2)=-5 and f(1)=4, which of the following statements could be false?
A) There exists c, where [-2,1], such that f(c)=0
B) There exists c, where (-2,1) such that f '(c)=0
C) There exists c where (-2,1) such that f(c)=3
D) there exists c where (-2,1) such that f '(c)=3
E) there exists c where (-2,1) such that f(c) is greater than or equal to f(x) for all x on the closed interval [-2,1]
I don't think that it is A or C because f(-2)=-5 and f(1)=4 so that means that there must be some values in between -2 and 1 that equal 0 and 3 since f is continuous right?
I know that it is not D because I used the mean value theorem and found that f '(c)=3
I don't think that it is E because it is a closed interval meaning that there is a max in [-2,1]
So i think that it is B because although there must be a max and min in [-2,1], they could be on the endpoints since it is a closed interval. The derivative of an endpoint wouldn't be 0
Is this right? Please help