We were given this question in class as a exercise to do at home but I am confused about it...
Let F(x,y) =(2x2-y)(x2-y).
1. Discuss the sign of F at various points of the plane by appropriate consideration of the regions into which the plane is divided by the two parabolas y=x2 and y=2x2.
2. Discuss the critical points of the function.
3. Show that along every straight line through the origin, the values of F reach a minimum at (0,0), but that F neither has a minimum or a maximum at (0,0).
So far I don't really know how to do part 1. As far as 2 goes I have calculated (0,0) to be the only critical point... Is this correct? As for 3 I can show that (0,0) is a saddle point but I don't understand how to show that along every line through the origin the values of F reach a minimum at (0,0)...
Any help is greatly appreciated.