calc homework, st

Question

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.

II. If f"(c) exists, then f"(c)≤0.

A. II Only

B. iii only

c. i and ii only

d i and iii only

e. ii and iii only.

Answer 1

If you think of the absolute value function it might help you to see that (I) is not necessarily true. You can have a relative maximum without the derivative existing at "c".

But if it DOES exist, then the derivative at x = c will be equal to zero so (II) is true.

The fact that you have a maximum at x = c (and that f'(c) = 0) does not mean that the second derivative is zero. In fact, if f'' exists and f' is continuous then f'' is negative when x = c is a maximum (f'' < 0 means that f(x) is concave down) so (III) is not true.

So the answer is A