An argument x0
at which f ' is 0, so that f itself is flat, is called a critical
point of f.
Quadratic functions all essentially look alike, particularly if you are willing to stand on your head. Their behavior, when centered about 0, is the behavior of ax2 + c. The constant c determines where it appears in its graph, but the look of the graph is determined entirely by the parameter a. If a is positive the function looks like a fatter or thinner x2; if a is negative it looks like a fat or skinny -x2. This tells us that f has a local minimum at x0 when its second derivative is positive just as x2 does, and has a local maximum when a is negative (f has a local maximum is a point at which it is as big or bigger than those in some open interval containing it.)
When a is zero, so that f and f 'both have critical points at x0, the quadratic approximation is flat and you must look to the cubic approximation to determine the behavior of f near that point.
Exercise 10.4 Under what circumstances will f have a maximum at x0 when both its first and second derivatives vanish there?