when is the particle speeding up and when is it slowing down Based on this graph i have to figuere out when the particle is speeding up and when it is slowing down. 
My understanding is that when velocity and accelaration have the same sign then we are speeding up. however if velocity and accelaration have different signs then we are slowing down. Using this concept i applied it to this problem and i came up
speeding up using interval notation.
$(0,1)and(3,4) $
I'm getting the answer wrong can someone please help me understand the missing piece to this puzzle.
Thanks
Miguel
You've got the right idea, but not the right execution. Let's call the direction of positive $x$ "up" and the opposed direction down.

It is moving upward--has a positive velocity--on $(0,1)$ and $(3,4)$. It is moving downward--has a negative velocity--on $(1,3).$ Put another way, the graph's height is increasing on $(0,1)$ as we move to the right--likewise in $(3,4)$--and decreasing on $(1,3)$ as we move to the right.

It is accelerating upward--has a positive acceleration--on $(2,4)$. It is accelerating downward--has a negative acceleration--on $(0,2)$. Put another way, the graph's slope is increasing on $(2,4)$ as we move to the right, and the graph's slope is decreasing on $(0,2)$ as we move to the right.

Now check where the signs match.
A particle usually speeds up when the velocity and the acceleration have the same signs. It slows down when the acceleration and velocity signs are different. My advice (even though it might take a while) would be to sketch the curve from the given function. Then, you can find where the particle speeds up and slows down.