Question

How do you maximize and minimize `f(x,y)=1/x+y^2+1/(xy)` constrained to `2<x/y<4`?

Answer 1

See below.

Explanation:

Using the Lagrange multipliers technique we can find the local minima/maxima. For this problem there are four local minima at

#( (f(x,y), x, y), (0.784703,-1.49595,-0.747977), (1.98053,1.84942,0.92471), (0.629311,-2.56756,-0.641891), (1.3392,3.0675,0.766876) )#

Attached the feasible region plot showing the local minima. The black arrow show the objective function growing rate at each point.