Lagrange multipliers with f(x,y) = xy²

[King_Silver]

Given a question like this:
Findhe maximum and minimum of [PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/eq0043M.gif[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/empty.gif subject to the constraint [PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/eq0044M.gif[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/empty.gif.

I know that 5 = 2λx, -3 = 2λy.

However, if I am given a question where f(x,y) = xy² how would I write this?
There is no sign separating the x and the y so I cannot simply presume what sign it is.

Would 1 = 2λx and 1 = 2λy be incorrect?
If so, why? and how is this sort of question dealt with when the x and y are being multiplied together instead of being subtracted? cheers

 

[BvU]

For the Lagrange method you look at the gradient of $f - \lambda g$.
So all you have to do is determine the partial derivatives $\partial f\over \partial x$ and $\partial f\over \partial y$ . These aren't constants any more in this case.