$$f(x,y)=\frac{2}{x^2+(y-1)^2+1}-\frac{1}{x^2+(y+1)^2+1} \tag{1}\label{1}$$ 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 $(0,\approx1.04)$.
Below are the calculations, in Mathematica:
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, instead of $f$ use $f\circ g$, where $g$ is a diffeomorphism of the plane moving the given three distinct points to the critical points of $f$.