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[GaimeGuy]

Let f be a function defined and continuous on the closed interval [a, b]. If f has a relative maximum at c and a < c < b , which of the following statements must be true?
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
(E) II and III only


I can't eliminate any of the three statements. f ' (c) must exist according to the mean value theorem, even without the information that it's a relative maximum.

The second derivative test states that a relative maximum at f(c) for a continuous function means that f(c) is concave down ( f '' (c) is < 0 ) or f(c) is an inflection point (f '' (c) would equal 0, in that case ).

Is the answer D, because f isn't necessarily defined as being differentiable at c?

 

[Suranga3]

I'm pretty sure the answer is C. Since the derivative at a point is just the slope at the point, and at point C there is a relative max. The slope there must be zero.

 

[psycho_snake]

Ive been working on maths for the last two days preparing for my exams. I just had to look at the work calculus and Ive got another headahce.

 

[GaimeGuy]

actually, now that I reread it, I think it's e.

What if you had -|x| as your equation? That's continuous. There would be a maximum at c = 0, but f ' (c) is nonexistant. So I is out.

If f(c) is a maximum, then f ' (c) = 0 or does not exist. Therefore, if f ' (c) exists, f ' (c) has to be 0. II is true.

If f(c) is a maximum, that means that f' is positive, but that f' is also decreasing at c or not changing position at all ( meaning that f '' (c) would be less than or equal to 0). III is true.

Yeah, I'm pretty sure it's E (II and III must be true). But thanks for your help