This concerns several problems in the Calculus 3 (multivariable Calculus) course, under minimization and maximization.

There are problems such as

Library/Michigan/Chap15Sec1/Q21.pg

Library/Michigan/Chap15Sec2/Q03.pg

(and maybe more) where a student is given a contour map (the level curves) of a function of two variables. Several critical points are plotted, and the student is asked to classify those points (local minimum, local maximum, saddle, etc.).

It is impossible to classify critical points in this manner. For instance, suppose that

G(x) = -17/96 * x^6 - 277/480 x^4 - 1/2 x^2 - 1/10 and f(x,y)=G(x^2+y^2).

Then f has the following properties:

* If you draw the contour map for f(x,y)=-1, you get a circle (radius 1).

* If you draw the contour map for f(x,y)=0, you get a circle (radius sqrt(2)).

* Contour maps for f(x,y)=R (where R is an integer) are bigger circles if R > 0, and nonexistent if R < -1.

This would indicate that you have a local minimum at (0,0); however,

* (0,0) is actually a local maximum.

It's clear what the intention is, and that the author(s) wanted to find an alternative way to give a "classify the critical points" problem (other than a formula for the original function), but this is not a proper way to do it, in my opinion. I feel that we should not be teaching techniques that don't work, and that these problems should be removed from the library.

What does everyone else think about this?