The given question is complete and makes logical sense. No further completion or correction is needed.
a) The curve is always concave and reaches its maximum height at (1,0).
b) The curve must be continuously differentiable and reach (1,0) without any sharp corners.
c) The curve must be symmetric with respect to the line y=0.5 and slope at (1,0) is 0.
d) The curve must minimize the integral of its square root.