Chapter 15.7, Problem 82E

Solitary critical points A function of one variable has the property that a local maximum (or minimum) occurring at the only critical point is also the absolute maximum (or minimum) (for example. f(x) = x²). Does the same result hold for a function of two variables? Show that the following functions have the property that they have a single local maximum (or minimum), occurring at the only critical point, but that the local maximum (or minimum) is not an absolute maximum (or minimum) on ¡².

a.     f(x, y) = 3xe^y – x³ – e^3y

b.     $f\left(x,y\right)=\left(2y^2-y^4\right)\left(e^x+\frac{1}{1+x^2}\right)-\frac{1}{1+x^2}$

This property has the following interpretation. Suppose that a surface has a single local minimum that is not the absolute minimum. Then water can be poured into the basin around the local minimum and the surface never overflows, even though there are points on the surface below the local minimum.