Counting Extreme Points

Let $f(x) = x - x^3\mathrm{e}^{ax+b}$, and suppose the tangent line to $y = f(x)$ at the point $(1, f(1))$ is $y = -x + 1$.

1. Find $a$ and $b$. 2. Let $g(x) = f'(x)$. Find the intervals on which $g(x)$ is monotonic. 3. Determine the number of extreme points of $f(x)$.