max u(X,Y)
s.t. XPx + YPy = m
Lagrangian
L(X,Y,t) = u(X,Y) + t(m - XPx - YPy)
FONC
dL/dX = du/dX - tPx = MUx - tPx = 0
dL/dY = du/dY - tPy = MUy - tPy = 0
which implies
MUx/Px = MUy/Py = t,or MUx/MUy = Px/Py
MUx = 2Y, MUy = 2X
MRS = MUx/MUy = Y/X
Px/Py = 2
so we have
Y/X = 2,or Y = 2X
with the other linear restriction: 4X + 2Y = 400
X = 50, Y = 100
U_max = 2*50*100 = 10k