What characteristics of the graph of a function can we discuss by using the concept of differentiation (first and second derivatives).
(1) The first derivative is used to find the intervals where a function is increasing or decreasing. If the 1st derivative is positive the function is increasing, negative it is decreasing on the interval.
Also, extrema of the function are located where the 1st derivative is zero or fails to exist. Thus we can find maxima/minima of a function. With knowledge of the behavior of a function (increasing/decreasing) we can also tell if the extrema are global (absolute maximum/minimum) or local.
(2) The second derivative tells the rate of increase/decrease. Graphically this is described by the concavity: a graph could be increasing and concave up like an exponential or increasing and concave down like a square root.
Concavity can also be used to tell if a point identified by the first derivative as an extrema is a maximum or minimum -- if the graph is concave up the point is a minimum; concave down then it is a maximum.
Points where the concavity changes are called inflection points and they can occur if the second derivative is zero.