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Another example

We slice a sphere of radius R by a cylinder of height h and radius R.  

Minimize the surface area of the cylindrical sides if the sphere is fixed.


Area of the cylindrical sides: A= 2prh

We minimize

Compure partial derivatives

The critcal condition becomes

But here it is the condition for maximum cylindric area; the minimum area, zero, occurs at h = 0, r = R and at r = 0, h = 2R.