Let $f(x, y, z) = xyz$
$h1(x, y, z) = x + y + z − 4$5
and $h2(x, y, z) = 2x − y$.
Minimize $f(x, y, z)$ subject to $h1(x, y, z) = 0$ and $h2(x, y, z) = 0$.
First part: Show that every feasible point is regular.
Clear since $h1$ and $h2$ will always be linearly independent given that $z$ is zero.
Use the first order necessary conditions to find all candidates for local minimum points.
Compute the tangent spaces to all the candidates.
Use second order necessary and sufficient conditions to decide which of the points are indeed local minimum points.
I obtained the following system with $\lambda_1$ = lambda one and $\lambda_2$ = lambda two:
$yz - \lambda_1 - 2\lambda_2 = 0 $
$xz - \lambda_1 - \lambda_2 = 0$
$xy - \lambda_1 = 0$
Is this system correct? Will answering 2 involve simply determining $x,y,z$ in terms of $L1/L2$ and solving? Is there a general procedure for answering parts 2 and 3? If nothing else, how might one compute tangent spaces for candidates?
Thank you for taking the time to read this rather long question.