Pay no attention to the other tutor's (non)answer.  You are to prove it
in general for any given value of the total surface area,  So we just
call it A.

-----------------------------------------

Since the problem mentions diameter rather than radius, let's
convert the standard formula for the volume of a sphere so that
it is in terms of the diameter D instead of the radius, so we
substitute  for r in the usual formula:



And also for the surface area of a sphere



















Substitute in:









Put that equal to zero:


Multiply both sides by 2 to clear the fraction:



Add the exponents of 6



Divide by   (we aren't interested in when D=0)





Square both sides:




Multiply both sides by 6 to clear the fraction:







Take positive square roots of both sides:



Now we only need to show that x also equals that.



We substitute for D:







Multiply top and bottom under the radical by 









So x = r and the proposition is true.

Edwin