Exercise 36

If a continuous function $f$ of a single variable has exactly one critical number with a relative maximum at that critical point, then the value of $f$ at that critical point is an absolute maximum. In this exercise we see that the same is not always true for functions of two variables. Let $f(x,y) = 3xe^y-x^3-e^{3y}$ (from “The Only Critical Point in Town” Test by Ira Rosenholz and Lowell Smylie in the Mathematics Magazine, VOL 58 NO 3 May 1985.). Show that $f$ has exactly one critical point, has a relative maximum value at that critical point, but that $f$ has no absolute maximum value. Use appropriate technology to draw the surface defined by $f$ to see graphically how this happens.