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

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Answer 1 · 2016-12-23T01:21:28

See below.

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.

enter image source here

How do you maximize and minimize f(x,y)=1/x+y^2+1/(xy)f(x,y)=1x+y2+1xyf(x,y)=1/x+y^2+1/(xy) constrained to 2<x/y<42<xy<42<x/y<4?