Applied optimization: Maximize the area of a rectangle inscribed in an ellipse

This complete solution solves the problem of finding the maximum area of a rectangle inscribed in the ellipse x^2/36 + y^2/144 = 1. Along the way, we will eliminate a variable from the objective function (area = 4xy) by using the equation for the ellipse. We'll also use the chain rule to take the derivative of the single-variable objective function. Lots of mistakes await us! Pitfalls! Subtle errors! And at the end, we'll arrive at a stunningly beautiful answer. Come along with me...

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