CURVE SKETCHING (GRAPHS) 0 105
In each of Problems 15.23 to 15-29, sketch the graph of a continuous function/satisfying the given conditions.
15.23 /(I) =-2, /'(1) = 0, f(x)>0 tor all x.
I Since f"(x)>0 for all x, the graph is concave upward. Since /'(1) = 0 and /"(1)>0, there is a
relative minimum at x = 1. Hence, the graph in Fig. 15-13 satisfies the requirements.
Fig. 15-13
Fig. 15-14
15.24
Fig. 15-15
I The graph must be concave upward for x<0 and concave downward for x>0, so there is an inflection
point at x = 0, y = 0. The line y = 1 is a horizontal asymptote to the right, and the line y = -l is a
horizontal asymptote to the left. Such a graph is shown in Fig. 15-16.
Fig. 15-16
15.26
/(0) = 0, /"(*)<0 for *>0, /"(x)>0 for x<0,
/(2) = 3, /'(2) = 0, /"(*)<0 for all x.
I Since f"(x)<0, the graph is concave downward. Since /'(2) = 0 and /"(2)<0, there is a relative
maximum at x = 2, y = 3. The graph in Fig. 15-14 satisfies these requirements.
15.25 /(!) = !, f"(x)<0 for *>1, f"(x)>0 for
I The graph is concave upward for x<\ and concave downward for x>l; therefore, it has an inflection
point at x = 1, y — 1- A possible graph is shown in Fig. 15.15.