Only Critical Point in Town Test

If there is only one Critical Point, it must be the Extremum for functions of one variable. There are exceptions for two variables, but none of degree $\leq 4$ . Such exceptions include

$\begin{displaymath} z=3xe^y-x^3-e^{3y} \end{displaymath}$

$\begin{displaymath} z=x^2(1+y)^3+y^2 \end{displaymath}$

$\begin{displaymath} z=\cases{ {xy(x^2-y^2)\over x^2+y^2} & for $(x, y)\not= (0, 0)$\cr 0 & for $(x, y)= (0, 0)$\cr} \end{displaymath}$

(Wagon 1991). This latter function has discontinuous $z_{xy}$ and $z_{yx}$ , and $z_{yx}(0,0)=1$ and $z_{xy}(0,0)=1$ .

References

Ash, A. M. and Sexton, H. ``A Surface with One Local Minimum.'' Math. Mag. 58, 147-149, 1985.

Calvert, B. and Vamanamurthy, M. K. ``Local and Global Extrema for Functions of Several Variables.'' J. Austral. Math. Soc. 29, 362-368, 1980.

Davies, R. Solution to Problem 1235. Math. Mag. 61, 59, 1988.

Wagon, S. ``Failure of the Only-Critical-Point-in-Town Test.'' §3.4 in Mathematica in Action. New York: W. H. Freeman, pp. 87-91 and 228, 1991.