Use Lagrange multipliers to prove that the rectangle with maximum area that has

Question

Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square. Let the sides of the rectangle be x and y and let f and g represent the area (A) and perimeter (p), respectively. Find the following. delta(x, y) = ? ?*delta*g = ?

Explanation / Answer

Let x and y be the dimensions of the sqaure. So, we want to maximize A(x,y) = xy subject to the constant g(x,y) = 2x + 2y = p. By Lagrange, grad A = ? * grad g ==> = ?. ==> y = 2? and x = 2? ==> y = x. Hence, we have a square. (Check: When x = y, 2x + 2x = p ==> x = p/4. So, A_max = p^2/16. This is maximal, by checking any other point satisfying g, like (x,y) = (0, p/2) which has A = 0.)

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