On Saturday, 26 October 2013 06:01:27 UTC+3, William Elliot wrote: > On Fri, 25 Oct 2013, firstname.lastname@example.org wrote: > > > > > Of course the proof boils down to elementary algebra, but I'm looking for a > > > way that avoids tedious calculations as much as possible (eg. some geometric > > > principle) ... > > > > What proof?
The proof I were to make substituting z as iy - x, and using Lagrange multipliers to solve the constraint. The original problem was not the one posted here. It was to prove that
(x*x + 2*x + y*y > 3) implies that ((x^3 - 3*x*y*y +1)^2 + (3*x*x*y-y*y*y)^2 > 1) for x ,y real numbers .
I've figured that if you write z = iy - x , you can rewrite it in a simple way, namely:
|z - 1| > 2 implies |z^3 - 1| > 1 The question is, does this now shortened problem have a short solution?
The shape described by |z^3 - 1| > 1 looks like a sort of cycloid (hypotrochoid ?), other than that and the standard procedures for solving a function subject to inequality constraints, I'm stumped as to what a proper proof might look like.