Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.

*f* (*x*) = *x*^{2}*e*^{x} - 4*e*^{x}

Answer

*f* (*x*) = *x*^{2}*e*^{x} - 4*e*^{x}

Rewrite *f* so we can tell what's going on.

*f* (*x*) = *e*^{x}(*x*^{2} - 4) = *e*^{x}(*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 (*x*^{2} + 2*x* - 4), which will be the only places where *f*' is zero.

The critical points are

and

The second derivative of *f* is

Again, we need to use the quadratic formula. We need to find the roots of (*x*^{2} + 4*x* - 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.