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10.4 Quadratic Behavior at Critical Points

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 a x 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?

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