I have a function with two constraints whose intersection is unitary circumference. $$x^2+y^2+z^2=1$$ and $$x+y+z=0$$ I can't understand why I cannot apply the lagrange multipliers method with only the intersection as constraints. When I have two constraints, I am restricting the domain of the function to the points that satify $g_1(X)=0$ and $g_2(X)=0$ so I can't understand why I just can't replace the two zero sets with the intersection set.
Could you help me? The correct lagrangian should be $$L(x,y,z,\lambda,\mu)=f(x,y,z)-\lambda (x^2+y^2+z^2-1)-\mu(x+y+z)$$ but I wish to understand why studying $$L(x,y,z,\lambda)=f(x,y,z)-\lambda (x^2+y^2-1)$$ is wrong. Thanks a lot
What's wrong is that $x^2 + y^2 - 1 = 0$ is not the intersection of your two constraints. In fact it has only two points in common with that intersection.