Rolle's Theoram

Question

Does there exist a function f such that f(0) = −1, f(2) = 6, and f '(x) ≤ 1

 for all x?

Yes

No    

Answer 1

If a function is continuous, then, although f'(x) can be <1 for some x (like this):

It CANNOT be <1 for ALL x otherwise you couldn't get from -1 to 6.

But, this problem doesn't say the function has to be continuous. In that case, what about this:

The derivative is always <1 but the problem is that the derivative isn't defined at x = 2 because of the jump.So you can't say f '(x) < 1 for all x.

So, it seems like the answer is "No"