I couldn't find a clever way so this is just straightforward.
We have a system of two conic equations. Find the solutions and pick the one that minimizes xy.
Solve both equations for y and substitute to get 1/(12-x)=3/8-1/x which simplifies to the quadratic 0=x^2-12x+32=(x-4)(x-8)
The solutions to the system (x,y) and their product:
(4,1/8) and xy=1/2
(8,1/4) and xy=2
So the smallest xy is 1/2.
Posted by Jer
on 2019-01-29 08:53:08