Calculating the control points of a cubic Bézier curve given extrema

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Is it possible to find the 2 missing control points of a Bézier curve given that:

  1. You know in advance that it's a cubic Bézier curve
  2. You know the endpoints
  3. You know the coordinates of the local extrema (max, min, and saddle points).

Also, if this info is not enough to define a unique curve, would you still be able to calculate all possible solutions for the control points?