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E) If A is countable, then A is closed. 4 z as a function defined on ޒ3 , subject to the constraint 53. What is the minimum value of the expression x x2 y2 (A) 0 z2 2? (B) (C) 2 (D) 34 (E) 35 5 2 54. The four shaded circles in Figure 1 above are congruent and each is tangent to the large circle and to two of the other shaded circles. Figure 2 is the result of replacing each of the shaded circles in Figure 1 by a figure that is geometrically similar to Figure 1. What is the ratio of the area of the shaded portion of Figure 2 to the area of the shaded portion of Figure 1 ?
Then x (1) By the Mean Value Theorem, there is a c1 in the interval 1, 2 such that f c1 f 2 2 f 1 1 f 2 f 1 0. (2) For each x 2, there is a cx in 2, x such that (3) For each x 2, (4) For each x 2, f x f x x f 2 2 f cx f 2 x f x x f 2 2 f cx . f c1 since f is strictly increasing. 2 f c1 . (5) lim f x x Which of the following statements is true? (A) The argument is valid. (B) The argument is not valid since the hypotheses of the Mean Value Theorem are not satisfied in (1) and (2). (C) The argument is not valid since (3) is not valid.
If K is compact, then every continuous real-valued function defined on K is bounded. II. If every continuous real-valued function defined on K is bounded, then K is compact. III. If K is compact, then K is connected. (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III 63. If f is the function defined by f x ÎÑ xe x2 x 2 Ï ÑÐ0 if x 0 if x 0, at how many values of x does the graph of f have a horizontal tangent line? (A) None (B) One Unauthorized copying or reuse of any part of this page is illegal.