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Lagrange multipliers question help please :)

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    Find the rectangle in the ellipse  x^2 + 4y^2 = 1 with the smallest perimeter.

    I know that if  L = f(x,y) + \lambda g(x,y) then  \lambda g(x.y) = \lambda(x^2 + 4y^2 - 1) .

    However in the solutions it says that  f(x,y) = 4x + 4y . Why isnt this  2x+2y for the perimeter of the rectangle? Or is this a mistake in the solution? :/

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    4x+4y is the perimeter of the rectangle.

    Think for a moment, the ellipse is one which is clearly centered at the origin.

    Hence, the length and breath of the rectangle would have to span both the -ve/+ve x and y axes in a symmetrical fashion to sit snugly within the ellipse.

    The horizontal side of the rectangle would be 2x.

    The vertical side of the rectangle would be 2y.

    Hence, perimeter would be 2* (2x+2y)= 4x+4y

    Hope this helps. Peace.


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Updated: May 23, 2012
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