Let f be a function defined and continuous on the closed interval [a,b]. If f has a realtive maximum at c, and a<c<b, which of the following statements must be true?

Question

I. f ' (c) exists

II. If f ' (c) exists then f ' (c)=0

III. If f ' (c) exists, then f '' (c) is less than or equal to 0

A: II Only

B: III Only

C: I and II only

D: I and III only

Answer 1

Let us examine all three statements.

(I): I don't think this holds. Consider this function on the interval [0,2]:

f(x) = x, x ∈ [0,1),

f(x) = 2 - x, x ∈ [1,2]

Clearly f has a local maximum at x = 1, but f'(1) does not exist (why?)

(II) True. A fundamental theorem in calculus gives us this, but an intuitive proof exists by simply drawing out the situation in graphical form (remember that differentiability implies continuity).

(III) Not true. Just because f'(c) exists does not imply that f''(c) does, much less so ensuring it attains a specific value.

Therefore only (II) holds.