EXAMPLE 6 Sketch the graph of f(x)- (A) The domain is R = (-00, oo) (B) The x- and y-intercepts are both (C) Since f-x) -x), fis odd (D) Since x2 + 2 is never 0, there is no vertical asymptote. Since f(x) â oo as x â oo and f(x) â-00 x2 2 Vand its graph is symmetric about the origin Video Exampleã® x â-o, there is no horizontal asymptote. But long division gives x2 2 f(x) - x - 1 2/x2 So the line y is a slant asymptote (x2 + 2)2 (x2 + 2)2 Since f'(x) > 0 for all x (except 0), f is increasing on (-co, co) (F) Although f(0)= (G) f"(x)= , f(x) does not change sign at 0, so there is no local maximum or minim 4x3 +12xx2 212- (x4+6x2) 2(x2 + 212x 4x(6x2 Since f"(x)=0 when x=0 or x=ty/6, we set up the following chart: Interval2x 6 x2(x2 + 2)3 F (x) CU on -, 6) CD on V6, 0) + CU on (0, 6) The points of inflection are (x, y) = (smaller x-value), (0, 0), and (larger x-value) (H) The graph is sketched in the figure