For the function, find all points of inflection or determine that no such points exist.
f (x) = xex
We find the derivatives using the product rule:
This is never undefined, and has its only root at x = -2. Since ex is always positive, it doesn't affect the sign of f". The only part that affects the sign of f" is the (2 + x):
f"(x) = ex(2 + x)
Since 2 + x is negative for x < -2 and positive for x > -2, the sign of f" changes at x = -2, so this is an inflection point. Here's the graph: