find a positive number such that the sum of its squared and its inverse is minimal

Answer 1

Let x = the positive number. "inverse" is not well defined (multiplicative inverse or additive inverse?). Assume additive or opposite.

So the sum is represented by y = x² + (-x).

Take the first derivative, set it equal to zero, convince yourself this generates a minimum.

Or, realize that the graph is a parabola, find the vertex. The x coordinate of the vertex is your number (assuming it is positive).Remember the axis of symmetry is given by x = -b/2a = 1/2.