How do you minimize and maximize f(x,y)=xe^x-yf(x,y)=xexy constrained to 0<x-y<10<xy<1?

1 Answer
Jun 5, 2016

There are two local minima points at p_1 = {0,1}p1={0,1} and p_2={0,0}p2={0,0}

Explanation:

We will searching for stationary points, qualifying then as local maxima/minima.

First we will transform the maxima/minima with inequality restrictions into an equivalent maxima/minima problem but now with equality restrictions.

To do that we will introduce the so called slack variables s_1s1 and s_2s2 such that the problem will read.

Maximize/minimize f(x,y) = x e^x - yf(x,y)=xexy
constrained to

{ (g_1(x,y,s_1)=x - y - s_1^2=0), (g_2(x,y,s_2)=x - y + s_2^2 - 1=0) :}

The lagrangian is given by

L(x,y,s_1,s_2,lambda_1,lambda_2) = f(x,y)+lambda_1 g_1(x,y,s_1)+lambda_2g_2(x,y,s_2)

The condition for stationary points is

grad L(x,y,s_1,s_2,lambda_1,lambda_2)=vec 0

so we get the conditions

{ (e^x + lambda_1 + lambda_2 + e^x x = 0), ( -1 - lambda_1 - lambda_2 = 0), (-s_1^2 + x - y = 0), ( -2 lambda_1 s_1 = 0), ( -1 + s_2^2 + x - y = 0), ( 2 lambda_2 s_2 = 0) :}

Solving for {x,y,s_1,s_2,lambda_1,lambda_2} we have

{(x =0, y = -1, lambda_1 = 0., s_1 = 1, lambda_2 = -1, s_2 = 0.), (x = 0, y = 0, lambda_1 =-1., s_1 = 0, lambda_2 = 0, s_2 = 1.) :}

so we have two points p_1={0,-1} and p_2 = {0,0}

Point p_1 activates restriction g_2(x,y,0) = 0,{lambda_2 ne 0, s_2 = 0} and point p_2 activates restriction g_1(x,y,0)=0,{lambda_1 ne 0, s_1 = 0}

p_1 is qualified with f_{g_2}(x) =1 + (e^x-1) x

and

p_2 is qualified with f_{g_1}(x) = (e^x-1) x

Computing

d/(dx)(f_{g_2}(0)) = 0

and

d^2/(dx^2)(f_{g_2}(0)) = 2

we conclude that p_1 local minimum point.

Analogously for p_2

d/(dx)(f_{g_1}(0)) = 0

and

d^2/(dx^2)(f_{g_1}(0)) = 2

so p_1,p_2 are local minima points

enter image source here