Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.
f (x) = x2ex - 4ex
rewrite f first
Rewrite f so we can tell what's going on.
f (x) = ex(x2 - 4) = ex(x - 2)(x + 2).
This is zero when x = ± 2, so the x-intercepts are (2,0) and (-2,0). The y-intercept is (0,-4).
The derivative of f is
We need to use the quadratic formula to find the roots of (x2 + 2x - 4), which will be the only places where f' is zero.
The critical points are
The second derivative of f is
Again, we need to use the quadratic formula. We need to find the roots of (x2 + 4x - 2), since these are the places f" will be zero.
Since the sign of f" does change at these x-values, these are both inflection points of f. We will find their full coordinates.
We need to cheat a little on the labeling, because the exact coordinates are so awful.