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the graph moves upward without bound on the left and the right. The graph is shown in Fig. 23-11.
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[CHAP. 23
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23.3 Sketch the graph of f (x) = x 4 8x 2 .
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As the function is even, we restrict attention to x 0. f (x) = 4x 3 16x = 4x(x 2 4) = 4x(x 2)(x + 2) f (x) = 12x 2 16 = 4(3x 2 4) = 12 x 2 4 3 2 = 12 x + 3 2 x 3
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The nonnegative critical numbers are x = 0 and x = 2. Calculating, we obtain: x 0 2 2 + 3 f (x) 0 16 80 9 16 32 0 f (x) rel. max. rel. min. in . pt.
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Checking the sign of f (x), we see that the graph will be concave downward for 0 < x < 2/ 3 and concave upward for x > 2/ 3. Because lim f (x) = + , the graph moves upward without bound on the right.
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The graph is sketched in Fig. 23-12. Observe that, on the set of all real numbers, f has an absolute minimum value of 16, assumed at x = 2, but no absolute maximum value.
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Fig. 23-12
Supplementary Problems
23.4 Determine the intervals where the graphs of the following functions are concave upward and the intervals where they are concave downward. Find all in ection points. GC Check your solutions with a graphing calculator. (a) f (x) = x 2 x + 12 (c) f (x) = x 3 + 15x 2 + 6x + 1 (b) f (x) = x 4 18x 3 + 120x 2 + x + 1 x (e) f (x) = 5x 4 x 5 (d) f (x) = 2x 1
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23.5 Find the critical numbers of the following functions and determine whether they yield relative maxima, relative minima, in ection points, or none of these. GC Check your solutions with a graphing calculator. (a) f (x) = 8 3x + x 2 (c) f (x) = x 3 5x 2 8x + 3 (b) f (x) = x 4 18x 2 + 9 (d) f (x) = x2 x 1 x2 (e) f (x) = 2 x +1
23.6 Sketch the graphs of the following functions, showing extrema (relative or absolute), in ection points, asymptotes, and behavior at in nity. GC Check your solutions with a graphing calculator. (a) f (x) = (x 2 1)3 (d) f (x) = x 4 + 4x 3 (g) f (x) = x 2 + 2 x (b) f (x) = x 3 2x 2 4x + 3 (e) f (x) = 3x 5 20x 3 (h) f (x) = x2 3 x3 (c) f (x) = x(x 2)2 (f ) f (x) = 3 x 1 (i) f (x) = (x 1)3 x2
23.7 If, for all x, f (x) > 0 and f (x) < 0, which of the curves in Fig. 23-13 could be part of the graph of f
Fig. 23-13
23.8 At which of the ve indicated points on the graph in Fig. 23-14 do y and y have the same sign 23.9 Let f (x) = ax 2 + bx + c, with a = 0. (a) How many relative extrema does f have (b) How many points of in ection does the graph of f have (c) What kind of curve is the graph of f 23.10 Let f be continuous for all x, with a relative maximum at ( 1, 4) and a relative minimum at (3, 2). Which of the following must be true (a) The graph of f has a point of in ection for some x in ( 1, 3). (b) The graph of f has a vertical asymptote. (c) The graph of f has a horizontal asymptote. (d) f (3) = 0. (e) The graph of f has a horizontal tangent line at x = 1. (f ) The graph of f intersects both the x-axis and the y-axis. (g) f has an absolute maximum on the set of all real numbers. 23.11 If f (x) = x 3 + 3x 2 + k has three distinct real roots, what are the bounds on k [Hint: Sketch the graph of f , using f and f . At how many points does the graph cross the x-axis ]
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