.Dear forum members,
in a problem I was asked to show that when the surface area of a fixed cylinder was at its minimum, the height of the cylinder would be equal to the diameter of the cross section of the cylinder.
I formed the equation for the surface area of the cylinder, and the differentiated it with respect to r(not sure why I choose r, though. Perhaps because it was the only factor that would influence both the area of the "bottoms" of the cylinder as well as the "body".).
A(r)=2pi*r^2 + 2 pi*r*h
A'(r)=4pi*r+2pi*h ... this derivative is incorrect, you cannot treat h as a constant.
solving for the stationary points of the derivative gives 2r=-h .
The result shows basically what I wanted to show, but the minus is probably not supposed to be there.
However, I noticed, that when I expressed h in terms of the volume and the radius
and then plugged that into the equation above and differentiated, the result was 2r=h.
... this is correct because V is a constant, and the variation of h is related to the variation of r.
Could someone explain to me why these two different ways of doing give a different result?