10.4 Quadratic Behavior at Critical Points

]> 10.4 Quadratic Behavior at Critical Points

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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?