]> 10.4 Quadratic Behavior at Critical Points

## 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?