Find the stationary points of $f$ of the minimization problem $$min_{x\in\mathbb{R}^2} 100(x_2 − x_1^2)^2 + (1 − x_1)^2$$ and determine which points are local and which global extremas.
Problem: I have only found one stationary point... It is $$\nabla f(x) =\begin{pmatrix} 400x_1^3-400x_2+2x_1-2\\200x_2-200x_1^2\end{pmatrix} \overset{!}{=} 0$$
The first equation is zero if $x_2=x_1^2$ and the second if $x_1=x_2$. So the only stationary point I get is $x=(1,1)$... is this local or global? Are there more stationary points? Where is my mistake? Thank you!!
There is no mistake. The point $(1,1)$ is the only stationary point. Now, since the objective function is differentiable in the open set $\mathbb{R}^2$, the maxima/minima can only occur in stationary points. Considering that $f(1,1)=0$ and that $f(x,y)\ge 0$, we conclude that this is a global minimum.
This is a well known example, the Rosenbrock function. It is used to test numerical optimisation algorithms precisely because there is a very flat region around the line $x_2=x_1^2$ that makes life very difficult for gradient based methods (and others).