Relationships between f, f prime and f double prime

when f'(x) is decreasing, f is

concave down; f'' is negative

when f'(x) is increasing, f is

concave up; f'' is positive

relative min of f occurs when f'..

crosses x axis (negative--positive)

relative max of f occurs when f'..

crosses x axis (positive--negative)

function decreasing when f'..

is negative (below x axis)

function is concave down when f''..

is negative (below x axis)

function increasing when f'..

is positive (above x axis)

function is concave up when f''..

is positive (above x axis)

function has a POI when f' is

a max/min OR when f' changes direction (inc-dec/dec-inc)