The sign (positive, zero, negative) of the first derivative tells us the direction of the graph of the original function.[br]If f '(a) > 0, then f is increasing at x = a.[br]If f '(a) < 0, then f is decreasing at x = a.[br]If f ' (a) = 0, then f has a horizontal tangent line at x = a.[br] If f '(a) = 0 and f "(a) >0, then (a, f(a)) is a local minimum.[br] If f '(a) = 0 and f "(a) < 0, then (a, f(a)) is a local maximum. [br] If f '(a) = 0 and f "(a) = 0, then (a, f(a)) might be an extremum or an inflection point.[br]If f has a local maximum at (a, f(a)), then either the graph is at an endpoint, f '(a) = 0, or f '(a) is undefined.

The second derivative tells us the concavity of the original function.[br]If f "(a) > 0, then f is concave up at x = a (and f ' is increasing at x = a).[br]If f "(a) < 0, then f is concave down at x = a (and f' is decreasing at x = a).[br]If f "(a) = 0 and f '(a) [math]\ne[/math] 0, then f has an inflection point at (a, f(a)) (and f' has an extremum at x = a).[br]If f "(a) = 0 and f' (a) = 0, then f with have either an inflection point or an extremum.[br]If f has an inflection point at (a, f(a)), then either f '(a) = 0 or f "(a) is undefined. [br]If f "(x) = 0 on an interval, then f is linear on that interval.