Prove no existing a smooth function satisfying ... related to Morse Theory i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ and $f(x_1)=f(x_2) = 0$. (This part is easy).
ii) Show that there does not exist a smooth function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$, s.t. $f(x,y) \geq 0$, $\forall (x,y) \in \mathbb{R}^2$, $f$ has exactly two critical points, $(x_1,y_1),(x_2,y_2)\in\mathbb{R}^2$ and $f(x_1,y_1)=f(x_2,y_2) = 0$. 
I have tried several methods, however, it does not work, could anybody help me out?

As required by "This is much healthier", I post a new thread to express my opinions as related to "user126154"'s answer, which is great, however there is something that I can't convince myself.
First of all, the compactness condition in the proof of "user126154" can be relaxed as proposed in Richard Palais: Topology Volume 2, Issue 4, 1963, which can be found here: 
Richard Palais's paper.
Secondly, let's consider the smooth function $h(x,y) = [(x+1)^2 + y^2][(x-1)^2 + y^2]$, it will be easily shown that the critical points are isolated, and the only two minimal are non-degenerate critical points.
Now, on the one hand, just as "user126154" proposed, we can constructed a new function the same way (the details are shown below):
Consider any diffeomorphism $\phi:\mathbb{R}^2 \rightarrow B$.
For the two minimal of $h$, $(-1,0);(1,0)$ denoted as $p_1,p_2$, now join $p_1$ to $p_2$ with a simple arc which avoids critical points other than the $p_1,p_2$. A regular neighborhood of such arc is a disc D where h has no critical points other than at $p_1$ and $p_2$. Now uses a diffeomorphism $\psi$ from B to D and compose with $h$, which is $h(\psi(\phi(x,y)))$. Then the constructed function should have only two critical points.
On the other hand, if the constructed function is exactly a counterexample, for the reason that the critical points of the constructed function are non-degenerate, we can apply the theorem in the paper above to show that there is at least another critical point.
So, what's the problem?
