Question

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

Answer 1

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.