How do you minimize and maximize #f(x,y)=x/y-xy# constrained to #0<x-y<1#?
Both choices respect the constraint.
There is a local máxima at
the local maximization/minimization problem with inequality restrictions is transformed into an equivalent one, now with equality restrictions, so we can apply the Lagrange Multipliers technique for its resolution. The lagrangian is
The stationary points are solutions of
Here we observe that
Also we have
This result shows that the point