MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Do there exist nonconstant real valued functions f $f$ and g $g$ such that the expression: $$f(x) -v/g(x)$$ Is is maximized at $x = v v$ for all positive real v?$v$?
Do there exist nonconstant real valued functions f and g such that the expression: $$f(x) -v/g(x)$$ Is maximized at x = v for all positive real v?