A Cubic and Its Chord

Let $f(x) = \dfrac{1}{3}x\left(x^2 + 3x + a\right)$.

1. Find the intervals on which $f(x)$ is monotonic, discussing all cases. 2. Suppose $f(x)$ has two extreme points $x_1, x_2$, and let $A(x_1, f(x_1))$, $B(x_2, f(x_2))$. Does there exist $a$ such that the intersection point of line $AB$ with the $x$-axis lies on the curve $y = f(x)$? If so, find $a$; if not, explain why.