How do you minimize and maximize f(x,y)=x2y3+xy2 constrained to 0<x+3xy<4?

1 Answer
Douglas K.
May 20, 2017

Use two Lagrange multipliers and two slack variables. Here is a reference regarding Inequality Constraints

Explanation:

Given: f(x,y)=x2y3+xy2

A little better form:

f(x,y)=x2y−3+xy2

We need to work on the constraint functions.

g1(x,y,s1)=x+3xy−s21=0
g2(x,y,s2)=4−x−3xy−s22=0

NOTE: s1ands2 are called "slack variables" they allow us to take up the slack for the inequality.

We have two constraint functions, therefore we have two Lagrange multipliers and the Lagrange function is:

L(x,y,λ1,λ2,s1,s2)=x2y−3+xy2+λ1x+3λ1xy−λ1s21+4λ2−λ2x−3λ2xy−λ2s22

The six partial derivatives are:

∂L(x,y,λ1,λ2,s1,s2)∂x=2xy−3+y2+λ1(1+3y)−λ2(1+3y)

∂L(x,y,λ1,λ2,s1,s2)∂y=−6xy−4+2xy+3x(λ1−λ2)

∂L(x,y,λ1,λ2,s1,s2)∂λ1=x+3xy−s21

∂L(x,y,λ1,λ2,s1,s2)∂λ2=4−x−3xy−s22

∂L(x,y,λ1,λ2,s1,s2)s1=−2λ1s1

∂L(x,y,λ1,λ2,s1,s2)s2=−2λ2s2

Set them equal to zero and solve them as a system of equations:

0=2xy−3+y2+λ1(1+3y)−λ2(1+3y)

0=−6xy−4+2xy+3x(λ1−λ2)

0=x+3xy−s21

0=4−x−3xy−s22

0=−2λ1s1

0=−2λ2s2

I substituted u,v,w,andz for λ1,λ2,s1ands2, respectively and gave the equations to WolframAlpha. It came up with:

x≈0.754146,y≈1.43467

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