The diagonals of a convex quadrilateral are mutually perpendicular. The sum of the lengths of the diagonals is 10. We want to find the maximum possible area of such a quadrilateral.
Let us denote by and the lengths of the two diagonals. Then the area of the quadrilateral is the following function of and :

If we solve for in terms of , we can reexpress this area as the following function of alone:

Thus we find that the maximum area is

You can earn partial credit on this problem.