From the information below, describe and then draw a rough sketch of the function around the point indicated.

(a) f (4) = 2, F(4) = 3, f “(4) = 5 With f(4) = 2, the function passes through the point (4, 2). With f′(4) = 3 > 0, the function is increasing; with f″(4) = 5 > 0, the function is convex. See Fig. 10-14(a).

(b) f (5) = 4, f′(5) = –6, f″ (5) = – 9 With f(5) = 4, the function passes through the point (5, 4). Since f″ (5) = –6, the function is decreasing; with f″(5) = –9, it is concave. See Fig. 10-14(b).

(c) f(4) = 5, f′(4) = 0, f″(4) = – 7 The function passes through the point (4, 5). From f(4) = 0, we know that the function is at a relative plateau at x = 4; and from f “(4) = –7, we know it is concave at x = 4. Hence it is at a relative maximum. See Fig. 10-14(c).

(d) f(2) = 4, f′(2) = – 5, f″(2) = 6 The function passes through the point (2, 4) where it is decreasing and convex. See Fig. 10-

14(d).

(e) f(3) = 2, f′(3) = 0, f″(3) = 8 The function passes through the point (3, 2), where it is at a relative plateau and convex. The function, therefore, is at a relative minimum at (3, 2). See Fig. 10-14(e).

(f) ,f(2) = 3, f′(2) = 7, f″(2) = –9 The function passes through the point (2, 3), where it is increasing and concave. See Fig. 10-
14(f).

Transcribed Image Text:

y (5,32) 32 28 24 20 16 12 00 8 4+ + -8 (3,0) 2 3 4 5 6 7 8 -12 -16 y= (x-3)5 -20 -24 -28 -32+ (1,-32)