I am solving $ grad [f(x,y,z)]$ = $\lambda$grad[g(x,y,z)]
I have then three equations, one involving x's and lambdas, another involving y's and lambdas and a third involving z's and lambdas.
I then write the constraint equation only in terms of lambda, using the relations of x,y,z to lambda from the knowledge of the above 3 equations. I can now solve for lambda.
My question is: now that I will plug in the known value of lambda into the 3 equations and solve for x, y, z, do I have to worry about satisfying this constraint again? Or, have I already done it, and now it's just a matter of getting my x's, y's and z's, and testing the function, f(x,y,z) to see what are the maximum and minimum values (subject to the constraint)?
Thanks,
Edit: Of course, once I get my x's and y's and z's, I can always check that these values satisfy the constraint equation, but I am looking more for a "guarantee", and that I understand a little more about how solving systems of (nonlinear) equations work.
Edit 2: Nevermind, it appears I still have to consider the constraint equation several more times.