I was just curious I see alot of examples on lagrange multipliers, but how would we set up a difficult or a general lagrange multiplier condition such that we want to minimise the surface area of a material used to cover up an object with a fixed volume $V$ in 3 dimensions?
Solving the problems I get, I just dont get how to do form constraint equations. Furthermore are there any interesting lagrange multiplier problems regarding optimising interesting shapes and volumes that give us an interesting result? Most problems in my book regarding optimising volumes seem very dull like a rectangular prism or a cone etc... are there some very complicated or mathematically interesting objects to optimise for surface area constraint to the volume?