The particle has positive acceleration when velocity is increasing. On the intervals (0, 1 s) and (6 s, 7 s), velocity is increasing--slope is positive--and so acceleration is positive.
Acceleration is a 2nd derivative, so you have to pay attention to where the graph holds water and spills water. Where it holds water, the acceleration is positive, and where it spills, it is negative. So it is slowing down, according to your definition, between $[0,1]$ and $[2,3]$, and speeding up otherwise. 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. Problem 5: Particle Motion. The position of a particle (in inches) moving along the x-axis after t seconds have elapsed is given by the following equation: s = f(t) = t4 – 2t3 – 6t2 + 9t. (a) Calculate the velocity of the particle at time t. (b) Compute the particle's velocity at t = 1, 2, and 4 seconds.  Positive Velocity and Negative Acceleration. Observe that the object below moves in the positive direction with a changing velocity. An object which moves in the positive direction has a positive velocity. If the object is slowing down then its acceleration vector is directed in the opposite direction as its motion (in this case, a negative acceleration).  Solution. a. The particle has positive acceleration when velocity is increasing. On the intervals (0, 1 s) and (6 s, 7 s), velocity is increasing--slope is positive--and so acceleration is positive. b. Acceleration is the slope of the v-t graph. The graph has zero slope whenever the graph is horizontal; that is, whenever the velocity is constant. This happens in the intervals (1 s, 2 s), 5 s, 6 s) and (7 s, 8 s). Displacement The displacement x of a particle is the changein its position:x x 2 x 1. (2-1)Displacement is a vector quantity. It is positive if the particle hasmoved in the positive direction of the axis and negative if the xparticle has moved in the negative direction. Average Velocity When a particle has moved from position x 1 to position x 2 during a time interval t t 2 t 1 ,its average velocity during that interval is(2-2)The algebraic sign of v avg indicates the direction of motion (v avg is avector quantity). 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. Negative Velocity and Positive Acceleration. Observe that the object below moves in the negative direction with a changing velocity. An object which moves in the negative direction has a negative velocity. If the object is slowing down then its acceleration vector is directed in the opposite direction as its motion (in this case, a positive acceleration). The dot diagram shows that each consecutive dot is not the same distance apart (i.e., a changing velocity). The velocity-time graph shows a line with a positive (upward) slope (meaning that there is a positive acceleration); the line is located in the negative region of the graph (corresponding to a negative velocity). That happens on this graph for t = 4.0 s. 2.12 A particle is moving with a velocity v o = 60.0 m/s in the positive x direction at t = 0. Between t = 0 and t = 15 s, the velocity decreases uniformly to zero. What is the average acceleration during this 15-s interval. 
http://math.stackexchange.com/questions/336419/when-is-the-particle-speeding-up-and-when-is-it-slowing-down http://www.calculus-help.com/5-particle-motion/ http://www.physicsclassroom.com/mmedia/kinema/pvna.cfm http://gauss.vaniercollege.qc.ca/pwiki/index.php/V-t_graphs http://www.cabrillo.edu/~jmccullough/physics4a/HW/HRW_ch02.pdf http://math.stackexchange.com/questions/336419/when-is-the-particle-speeding-up-and-when-is-it-slowing-down http://www.physicsclassroom.com/mmedia/kinema/nvpa.cfm http://ux1.eiu.edu/~cfadd/1350/Hmwk/Ch02/Ch2.html
when does the particle have positive acceleration