# If there were more than two points such that f x 0

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= 0. If there were more than two points such that f ( x ) = 0, then by Rolle’s theorem, there must exist a point c such that f 0 ( c ) = 0. Since none exists, this means that there cannot be more than two points where f ( x ) = 0. Since we have already found two such points, this means there are exactly two points where f ( x ) = 0. 7. (a) We simply compute the derivatives of the abscissa and ordinate with respect to time and put them equal to each other. i.e. dy dt = 2 x dx dt = dx dt Hence x ( t ) = 1 / 2. (b) If x = sin t and y = sin 2 t , then dx dt = cos t and dy dt = 2 sin t cos t . This gives us, cos t (2 sin t - 1) = 0 t = π 2 , π 6 Hence, the rates are given by, dy dt = dx dt = 0 dy dt = dx dt = p 3 / 2 4
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