 Solution

Write $f(x,y)=g(x)+g(y)$ with $g(s)=s^3/12-s$. On $[0,4]$, its derivative is $g^{\prime }(s)=s^2/4-1$, negative below two and positive above two. Compare the critical and endpoint values:Each coordinate can be chosen independently. Therefore the global minimum is $-8/3$ at $(2,2)$, and the global maximum is $8/3$ at $(4,4)$. This comparison includes the boundary of the rectangle, rather than classifying only interior stationary points.