Find the $x$ at which the local maxima of a function ocours. 
The function $f(x) = \int\limits_{-1}^{x}t(e^t-1)(t-1)(t-2)^3(t-3)^5 dt$ has a local maxima at $x=?$

First, I differentiated $f(x)$ and found its roots.
That came out to be $x = 0,1,2,3$. Now, one of those numbers, when plugged into $f(x)$, must give the largest value compared to the others.
In our case, since we are integrating, we would need to find the $x$ that minimizes the negative area of the function that we are integrating.
By further work and analyzing the function, I figured out that the answer must be $0$ or $2.$
But, I don't know what to do next?
Any help would be appreciated. 

HINT:
The sign of $$f^{'}(x)=x(e^x-1)(x-1)(x-2)^3(x-3)^5$$  is $-,+,-,+$ respectively on the intervals $(-1,1),(1,2),(2,3),(3,\infty).$ From this you can finish without further calculation.
