Let p(x,y) and q(x,y) be polynomials in two variables such that for x≥0,y≥0 the following conditions hold:
(i)p(x,y) and q(x,y) are increasing functions of x for every fixed y.
(ii)p(x,y) is an increasing and q(x) is a decreasing function of y for every fixed x.
(iii)p(x,0)=q(x,0) for every x and p(0,0)=0.
Show that the simultaneous equations p(x,y)=a,q(x,y)=b have a unique solution in the set x≥0,y≥0 for all a,b satisfying 0≤b≤a but lack a solution in the same set if a<b. functioninequalitiesalgebrapolynomialalgebra unsolved