Letf (x)andg(y)have local minima at x = a and y=b respectively. Show that f (x)+g(y) has aminimum at(a,b). Deduce that there exists a neighbourhood of(a,b) in which all solutions of the family of...

Letf (x)andg(y)have local minima at x = a and y=b respectively. Show that f (x)+g(y) has aminimum at(a,b). Deduce that there exists a neighbourhood of(a,b) in which all solutions of the family of equations
 represent closed curves surrounding(a,b).

Show that (0, 0) is a centre for the system
 and that all paths are closed curves.