Hi. I still need help with a constrained optimization problem. A couple days ago I posted a similar problem. MHF Member "Anonymous" helped me to get farther along, but now I'm stuck again. Here's the problem and how far I've gotten:

Maximize

subject to the constraint

I.e.

The Lagrangian is

Here are the first order conditions:

My confusion is that I see two solutions to this system of equations. First:

Setting all of the equations equal to each other...

and then simplifying gives us...

The second solution comes when you solve for

in one of the equations and then substitute this in for lambda in one of the other equations...

Solving for lambda in equation three:

Substituting this in for lambda in equation one:

Finally, setting this equal to equation two and simplifying:

So, on the one hand I get

while on the other hand the same system of equations simplifies to

. What gives? Do I have to now set these two equations equal to each other or what?