I do not like this approach. Because this only tells you if have a local minimum.
We have that and .
Thus, you need to minimize on the surface in the first octant bounded by the pane . By Extreme Value Theorem there will exists a minimimum now use partial derivatives to find all the critical points and just compute their values to see which one is the smallest one (no 2nd partials test here). But the problem is you also need to check the boundary of this surface which is going to take a lot of time.
Have you tried doing this without any calculus let be sides of the triangle then and so on. And proving it from there?