$$f(x,y)=\frac{2}{x^2+(y-1)^2+1}-\frac{1}{x^2+(y+1)^2+1} \tag{1}\label{1}$$
will do. 

Indeed, this function $f$ has exactly three critical points: a saddle point $(0,\approx-8.54)$, a point of a local minimum at $(0,\approx-1.14)$, and a point of a local maximum at $(0,\approx1.04)$. 

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For any given three distinct points on the plane to be 
exactly the critical points (again, a saddle, a local minimum, and a local maximum) of a function, use $f\circ g$ instead of $f$, where $g$ is a diffeomorphism of the plane moving the given three distinct points to the critical points of $f$. 

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Below are the calculations for $f$, in Mathematica:

[![enter image description here][1]][1]
[![enter image description here][2]][2]

 

  [1]: https://i.sstatic.net/LAsPA.png
  [2]: https://i.sstatic.net/P8cgD.png