f is increasing

positive

f is decreasing

negative

f has a local max

f' changes from + to -

0 or dne

if f'x=0 then f''x=-

f has a local min

f' changes from - to +

0 or dne

f is concave up

f' is increasing

f is concave down

f' is decreasing

f has an inflection point

f'(x) has relative max/min