Home  18.013A  Chapter 10 


An argument x_{0} at which f ' is 0, so that f itself is flat, is called a critical point of f.
When f " is not zero at such a point, its quadratic approximation there is a quadratic centered about x_{0}.
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 ax^{2} + 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 x^{2}; if a is negative it looks like a fat or skinny x^{2}. This tells us that f has a local minimum at x_{0} when its second derivative is positive just as x^{2} does, and has a local maximum when a is negative (f has a local maximum at 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 x_{0}, the quadratic approximation is flat and you must look to the cubic or higher approximation to determine the behavior of f near that point.
Exercise 10.4 Under what circumstances will f have a maximum at x_{0} when both its first and second derivatives vanish there?
