Answer
f (x) = x2ex – 4ex
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
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_52.png)
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.
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_53.png)
The critical points are
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_53a.png)
and
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_53b.png)
The second derivative of f is
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_54.png)
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.
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_55.png)
Since the sign of f" does change at these x-values, these are both inflection points of f. We will find their full coordinates.
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_56.png)
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_57.png)
We need to cheat a little on the labeling, because the exact coordinates are so awful.