3. (2001 National College Entrance Examination for Two Provinces and One City) Given the function $f(x)=x^{3}-3 a x^{2}+2 b x$ has a local minimum value of -1 at the point $x=1$, try to determine the values of $a$ and $b$, and find the monotonic intervals of $f(x)$.
3. From the given, we have
$$
f(1)=1-3 a+2 b=-1
$$

Also, $f^{\prime}(x)=3 x^{2}-6 a x+2 b$,
thus $f^{\prime}(1)=3-6 a+2 b=0$.

From (1) and (2) combined, we get $\left\{\begin{array}{l}3 a-2 b=2, \\ 6 a-2 b=3 .\end{array}\right.$ Therefore, $\left\{\begin{array}{l}a=\frac{1}{3}, \\ b=-\frac{1}{2} \text {. }\end{array}\right.$ Hence, the function is $f(x)=x^{3}-x^{2}-x$. This gives $f^{\prime}(x)=3 x^{2}-2 x-1$.
According to the properties of quadratic functions, when $x1$, $f^{\prime}(x)>0$; when $-\frac{1}{3}<x<1$, $f^{\prime}(x)<0$.
Therefore, the function $f(x)$ is increasing on the intervals $\left(-\infty,-\frac{1}{3}\right)$ and $(1,+\infty)$; and decreasing on the interval $\left(-\frac{1}{3}, 1\right)$.