How do you minimize and maximize f(x,y)=x+y constrained to 0<x+3y<2?

1 Answer
Cesareo R.
Aug 1, 2016

This problem is unbounded.

Explanation:

The linear function to maximize/minimize f(x,y) grows
in the direction of its gradient

grad_f = grad f(x,y) = {(partial f)/(partial x),(partial f)/(partial y)}= {1,1}

the linear restrictions offer boundaries at

g_1(x,y) = x+3y = 0
g_2(x,y)=x+3y=2

with constant declivity given by the vector

vec r_1 = vec r_2 = vec r = {3,-1}. Note that the declivity vector is normal to the restriction gradient vector given by

grad_{g_1} = grad_{g_1} = grad_g = {1,3}

Concluding, the projection of grad_f onto vec r is

<< grad_f, vec r >>/norm(vec r)=<< {1,1},{3,-1} >>/norm({-3,1}) = 2/sqrt(10) = C^{te} so f(x,y) keeps growing or decreasing along those boundaries.

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