Many things about increasing/decreasing/max/min and $f'$
are rather delicate: it's very easy to confuse statements,
definitions, etc.

If you have a -picture- in mind
(preferably a few pictures, chief among them
$\abs{x}$, $x^2$ and $x^3$ about $0$)
then it's very easy to reproduce them, but it's -hard- to memorize all
of them.
EXAM!! -- having a large number of these (say, True/False)
  on an exam would be great: there's no way to cram them all in.

EG:
if $f$ is increasing, is $f'(x)>0$ for all $x$?
  no: $x^3$
if $f'>0$ for all $x$, is $f$ is increasing?
  yes: FTC, or MVT
if $f$ has a global maximum at $x$, is $f'(x)=0$?
  no: needn't be differentiable
give an example of a non-decreasing function that is not increasing
  (flat for a while)
