EXAMPLE 3 Sketch the graph of f(x) = 4xe^x.
(A) The domain of f is R
(B) The x- and y-intercepts are both 0.
(C) Symmetry: None
(D) Because both 4x and e^x become large as x -> infinity, we have the limit of 4xe^x as x -> infinity is infinity. As x -> -infinity, however, e^x -> and so we have an indeterminate product that requires the use of l'Hospital's Rule:
lim x->-infinity 4xe^x = lim x->-infinity 4x / e^-x = lim x->-infinity 4 / -e^-x = lim x->-infinity -4e^x =
Thus the x-axis is a horizontal asymptote.
(E) f '(x) = 4xe^x + 4e^x =
Since e^x is always positive, we see that f '(x) > 0 when 1 + x > 0, and f '(x) < 0 when 1 + x < 0. So f is increasing on ( , infinity) and decreasing on (-infinity, ).
(F) Because f '(-1) = and f ' changes from negative to positive at x = , f(-1) = -4e^-1 is a local (and absolute) minimum.
(G) f ''(x) = (4 + 4x)e^x + 4e^x =
Since f ''(x) > 0 if x > and f ''(x) < 0 if x < , f is concave upward on ( , infinity) and concave downward on (-infinity, ). The inflection point is (-2, ).
(H) We use this information to sketch the curve in the figure.