Question

The function f is continuous for -2≤0≤2 and f(-2)=f(2)=0. If there is no c, where -2≤c≤2, for which f ' (c) = 0, which of the following must be true?

a for -2≤k≤2, f ' (k) > 0
b for -2≤k≤2, f ' (k) < 0
c for -2≤k≤2, f ' (k) exists
d for -2≤k≤2, f ' (k) exists, but f ' is not continuous
e For some k, where -2≤k≤2, f ' (k) does not exist

Answer 1

Rolle's Theorem states:

Suppose that f is continuous on a closed interval, that the derivative f' exists at every point of the open interval, and that f(a)=f(b)=0. Then there exists at least one point c in the open interval such that f'(c)=0.

Which part of the hypothesis could be false?

Answer 2

I don't think I ever learned Rolle's Theorem, but the answer is supposed to be E. I'm not sure how it is supposed to be though.

Answer 3

What about the Mean Value Theorem? Rolle's Theorem is just a special case of that. Look at what needs to happen to use the theorem:

1) f is continuous on a closed interval [a,b]

2) the derivative f' exists at every point in the open interval (a,b)

3) f(a)=f(b)=0.

One of these must not be true if your function doesn't have a point c such that f'(c)=0.