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# 11.28 - Section 4.5 concluded Example 5 Find the area of...

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Section 4.5, concluded: Example 5: Find the area of the largest rectangle that can be inscribed in a semicircle of radius r . Maximize A = 2 xy subject to x 2 + y 2 = r 2 with x , y 0. We used symmetry last time to argue that the optimum solution should have x = y ; let’s do a rigorous analysis, without using symmetry arguments. Use implicit differentiation: A = 2 y + 2 xy 2 x + 2 yy = 0 y = – x / y (undefined when y = 0) Critical points: A undefined y undefined x = r , y =0 (an endpoint) A = 0 0 = 2 y + 2 x (– x / y ) = 2 y – 2 x 2 / y 2 y = 2 x 2 / y y 2 = x 2 y = x x = y = r /sqrt(2) What kind of critical point is the latter? A ′′ = 2 y + 2 y + 2 xy ′′ 2 + 2 y 2 + 2 yy ′′ = 0 y ′′ = (–1– y 2 )/ y y ′′ = (–1– x 2 / y 2 )/ y y ′′ = – 1/ y x 2 / y 3

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A ′′ = 4 y + 2 xy ′′ = 4(– x / y ) + 2 x (– 1/ y x 2 / y 3 ) = – 4 x / y – 2 x / y – 2 x 3 / y
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11.28 - Section 4.5 concluded Example 5 Find the area of...

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