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HW4 Solutions - ASSIGNMENT 4 SOLUTIONS Problem 1...

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ASSIGNMENT 4 - SOLUTIONS Problem 1. Differentiating with respect to x we get 2 3 x - 1 / 3 + 2 3 y - 1 / 3 y = 0 . It follows that y = - 3 y/x . Problem 2. To find the points where the tangent to a curve is horizontal, we solve y = 0. Take y 2 = x 3 + x . Differentiating with respect to x we get 2 yy = 3 x 2 + 1, so: y = 3 x 2 + 1 2 y Then y = 0 precisely when 3 x 2 + 1 = 0, that is, never. In this case, the number of points is 0. Take y 2 = x 3 - x . Differentiating with respect to x we get 2 yy = 3 x 2 - 1, so: y = 3 x 2 - 1 2 y Then y = 0 precisely when 3 x 2 - 1 = 0, that is, when x = ± 1 3 . Plugging x = 1 3 in the original relation y 2 = x 3 - x , we get y 2 = - 2 3 3 which has no solutions. Plugging x = - 1 3 in the original relation y 2 = x 3 - x , we get y 2 = 2 3 3 which has two solutions. In this case, the number of points is 2. Take y 2 = x 3 - x + 1. Differentiating with respect to x we get 2 yy = 3 x 2 - 1, so: y = 3 x 2 - 1 2 y Then y = 0 precisely when 3 x 2 - 1 = 0, that is, when x = ± 1 3 . Plugging x = 1 3 in the original relation y 2 = x 3 - x + 1, we get y 2 = - 2 3 3 + 1 which has two solutions.
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