R Video Example EXAMPLE 4 (1) (11) (III) Sketch a possible graph of a function f that satisfies the following conditions. f'(x) > 0 on (-, 4), f'(x) < 0 on (4,0) f"(x) > 0 on (-, -8) and (8, ), f"(x) < 0 on (-8, 8) lim f(x) = -3, lim f(x) = 0 X118 840 SOLUTION Condition (1) tells us that f is increasing on (-, 4) and decreasing on (4,0). Condition (ii) says that fis concave upward on ,), and concave downward on ). From (iii) we know that the graph of f has two horizontal asymptotes: y = -0. and y = 0. and First we draw the horizontal asymptote y = as a dashed line (see the figure). We then draw the graph of f approaching this asymptote at the far left, increasing to its maximum point at x = 4 and decreasing toward the x-axis at the far right, We also make sure that the graph has inflection points when x = -8 and Notice that we made the curve bend upward for x < -8 and x > 8, and bend downward when x is between and 8