Inequalities problem?

Question

prove that:

[1/(1 - x²)] + [1/(1 - y²)] ≥ 2/(1 - xy)

given that x and y are positive real numbers.

Answer 1

♠ consider function
z(x,y)=1/(1 -x^2) +1/(1 -y^2) -2/(1-xy);
and consider ∂z/∂y or ∂z/∂x; no matter which as x and y are symmetrical in z(x,y);
♣ now let ∂z/∂y=0 for max/min analysis;
here you’ll see that y=x, that proves z(x,x) = 0;
further analysis proves that y=x situation means min of z(x,y);
♥ thus for y≠x we get z(x,y) > 0;
☺whiz-kids like you feel offended big time when the whole work is shown; u r welcum 2 clik me tho if still stuck;