The Brachistochrone Is The Cycloid
Given two points A and B, with A not lower than B, there is just one upside down cycloid that passes through A with infinite slope, passes also through B and does not have maximum points between A and B. This particular inverted cycloid is a brachistochrone curve. The curve does not depend on the body's mass or on the strength of the gravitational constant.
The problem can be solved with the tools from the calculus of variations.
If the body is given an initial velocity at A, or if friction is taken into account, the curve that minimizes time will differ from the one described above.
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