Question

Consider the following optimization problem:
optimize z(x, y) = 2xy
subject to x^2 + xy + 4y^2 = 40
x ≥ 0 and y ≥ 0.

This is an exam question and i did it wrong.
basically, the Lagrange criteria says this is not feasible but i dont understand why

Answer 1

\[z=2xy \\ x^2+xy+4y^2=40 \\z_x=\lambda g_x \\ z_y=\lambda g_y \\ x^2+xy+4y^2=40\]
so we want to solve the bottom 3 equations
like I'm just trying to see if it does look not feasible

Answer 2

\[2y= \lambda (2x+y) \\ 2x=\lambda(x+8y) \\ x^2+xy+4y^2=40\]
so you played with these 3 equations right?

Answer 3

I actually think it works if I didn't make a mistake.

Answer 4

I'm sure you got somewhere that
\[8y^2=2x^2 \\ \text{ or } 4y^2=x^2 \]
plug that into your constraint

Answer 5

are you there?

Answer 6

this also might help
since y>=0
then y|y| can be written as y^2

Answer 7

yes i got that, but could u explain why the points are not feasible to start with?

Answer 8

What do you mean by feasible? I was able to obtain point through lagrange

Answer 9

\[4y^2+2y^2+4y^2=40 \text{solve for y then x comes pretty easily afterwards} \]