Home  18.013A  Chapter 10 


An argument ${x}_{0}$ at which $f\text{'}$ 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 $f\text{'}$ 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?
