Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.
f (x) = (1-x)ex
The y-intercept is (0,1). The x-intercept is (1,0).
The derivative is
f'(x) = (-1)ex + (1-x)ex = -xex.
This is zero when x = 0, so (0,1) is a critical point as well as an intercept.
The second derivative is
f"(x) = -ex-xex = -ex(1 + x).
This is zero when x = -1 and changes sign at x = -1, so we have an inflection point at
For the sake of graphing, it's useful to know that