Brain Teaser - Minimum distance between two curves

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The my millimeterpaper and compas gives solution of 0.35 and 0.57 with scale of 10. Interesting, I definedly need to make some rereading of my old mathbooks.

Interesting teaser indeed. The tangent way is not solid even at this case, ie. points x=1/4 and x=1/2 have same angle and because of that the angle of normals are same in multitude of places, but like you CR Haeger do mention normal which is line and is normal for both curves and have same zero for both curves is the solution. Another solution ofcourse would be a circle which center point is one or another of the curves and have radious where there is only one common point for another curve. Now I should just figure out how to construct a function for it, no idea.

Hmm... since derivative of given function is angle of tangent and we know that angle of normal is -(a^-1) so the function for a normal of given function is therefore -1/f'(x) and we know normal of both function is the same we would assume that we find the right solution where both normal functions intersects, but since we also know that there is more than one solution for angle of tangents so there is more than one solution for angle of normals we need more glue, so I need to yet vacuumclean my dustbin to figure out how the line function at given point were build. Oh this is fun in bizarre way, have been missing it kind of.