10.4 Quadratic Behavior at Critical Points

An argument x₀ 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 x₀.

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 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 x₀, 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 x₀ when both its first and second derivatives vanish there?