Show uniqueness of a point

Question

I use the banana function $F(x_1,x_2)=(1-x_1)^2+100(x_2-x_1^2)^2$ and I found the minimum point X to be (1,1).

I need to show the uniqueness of that point. Could you please help me on how to show this?

Answer 1

I'll give some hints:

First show that the point is a local minimum.

Then prove that the function is strictly convex. (For both points you could use properties of the functions hessian matrix, if you are allowd to do so - but of course these are not the only ways.)

This implies that your local minimum is also a global minimum.