This exercise shows you how to determine the coordinates

Chapter 4, Problem 46

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This exercise shows you how to determine the coordinates of the lowest point on the middle branch of the curve in Figure 17. The basic idea is as follows. Suppose that the required coordinates are (h, k). Then the horizontal line y k is the unique horizontal line intersecting the curve in one and only one point. (Any other horizontal line intersects the curve either in two points or not at all.) In steps (a) through (c) that follow, we use these observations to determine the point (h, k). (a) Given any horizontal line y k, its intersection with the curve in Figure 17 is determined by solving the following pair of simultaneous equations: In the second equation of the system, replace y with k and show that the resulting equation can be written (1) (b) If k is indeed the required y-coordinate, then equation (1) must have exactly one real solution. Set the discriminant of the quadratic equation equal to zero to obtain and deduce from this that k 1 or k 25/9. The solution k 1 can be discarded. (To see why, look at Figure 17.) (c) Using the value y 25/9, show that the corresponding x-coordinate is 1/2. Thus, the required point is 47.