Exercise 35

If a continuous function $f$ of a single variable has two critical numbers $c_1$ and $c_2$ at which $f$ has relative maximum values, then $f$ must have another critical number $c_3$, because “it is impossible to have two mountains without some sort of valley in between. The other critical point can be a saddle point (a pass between the mountains) or a local minimum (a true valley).” (From Calculus in Vector Spaces by Lawrence J. Corwin and Robert H. Szczarb.) Consider the function $f$ defined by $f(x,y) = 4x^2e^y -2x^4 -e^{4y}$. (From Ira Rosenholz in the Problems Section of the Mathematics Magazine, Vol. 60 NO. 1, February 1987.) Show that $f$ has exactly two critical points, and that $f$ has relative maximum values at each of these critical points. Explain how this function $f$ illustrates that it really is possible to have two mountains without some sort of valley in between. Use appropriate technology to draw the surface defined by $f$ to see graphically how this happens.