How to solve such an optimization problem efficiently？?

Hey!! Need help with an optimization problem. Don't know how to solve for y.?

• Find two-non-negative numbers of x & y such that x^2+y=1 and xy=max.

• Answer:

  From the first equation we have that y=1-x^2 and therefore x has to be between -1 and 1 for y to be non-negative and, of course, as x itself has to be non-negative, x has to lie between 0 and 1. The objective to optimise is V = xy = x (1-x^2) = x - x^3 so dV/dx = 1 - 3 x^2 which is zero for x = 1/√3 (the other solution, -1/√3 is outside of the permissible bounds) Then y = 1 - (1/√3) so, performing the calculations, x=0.57735 and y = 0.42265