Expert: Scotto - 11/25/2011

Question
Dear Professor, I am an Italian boy with a school and university curriculum in humanities, who is trying to remediate some of his ignorance in natural and formal sciences. I have been studying university scientific textbooks by myself since last year, beginning with an institutions of mathematics book (general mathematics, used at biology), etc., and am trying to improve my knowledge of mathematics by following a calculus book now.
My books give several statements about convexity and increasing/decreasing character of functions and their derivatives, which I have tried to elaborate into the following ideas. Are they right?
I would say that, on a given interval (I call f a function and f' its derivative function):
-f is strictly convex if and only if f' is strictly increasing;
-f is strictly concave if and only if f' is strictly decreasing;
-f is non-strictly convex if and only if f' is non-strictly increasing;
-f is non-strictly concave if and only if f' is non-strictly decreasing;
I am asking you because my book is not always systematical in giving these definitions for each case, and I would like to know whether the double implications are correct. Hoping that you can help me, I thank you with all my heart.
Yours sincerely,
Davide

Answer
-f is strictly convex if and only if f' is strictly increasing;

Note that f'(x) is also the slope of f(x).  If f'(x) is strictly increasing,
that means the slope is ever increasing, which makes f(x) concave up.

-f is strictly concave if and only if f' is strictly decreasing;

If f'(x) is strictly decreasing, that mean the slope is constantly decreasing,
and this makes f(x) concave.

-f is non-strictly convex if and only if f' is non-strictly increasing;

If f'(x) is non-strictly increasing over some interval, then since f'(x) is the slope,
that means over the same interval, f(x) is non-strictly convex.  If f'(x) is 0 at some point,
f(x) is a straight line at that point.  If f'(x) is decreasing over some interval,
then f(x) is concave over that interval.

-f is non-strictly concave if and only if f' is non-strictly decreasing;

If f'(x) is 0 over some interval, f(x) is not concave, for it is a line over that interval.  If f'(x) is increasing over some interval, then over the same interval, f(x) must be convex.