Isoperimetric problem

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Among all closed plane curves with a given perimeter find one that maximizes the area. This is also known as Queen Dido's problem and serves as a classical problem for variational calculus. Its significance in mathematical programming is that it led to Lagrange's multiplier theorem. Restricting shapes to rectangles, the problem is:

LaTeX: 
\max \{xy : x+y=p, \, x \ge 0, \, y \ge 0\},

where LaTeX: 2p is the perimeter. For positive LaTeX: x and LaTeX: y and multiplier LaTeX: u, Lagrange's multiplier conditions requires LaTeX: y-u=0 and LaTeX: x-u=0, so LaTeX: x=y, which means the solution is the square.


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