# Finding change in speed from x vs t graph

1. The problem statement, all variables and given/known data

For a particle moving in one dimension, x vs t graph is given below. At each point state whether speed is increasing, decreasing or not changing.

2. Relevant equations

Just the basic relation between position, velocity (first derivative of position) and acceleration (second derivative of position).
Also, when this graph is concave up (curved upwards) the acceleration is positive. And if concave down (curved downwards) the acceleration is negative.
When v and a have same sign the speed is increasing; if v and a have opposite sign speed is decreasing.

3. The attempt at a solution

For point P:
Speed is not changing because there is no curvature at that point (ie acceleration is zero)

For point Q:
The curve is concave down so a<0 and from the graph the slope of tangent line at Q is zero so v=0

For point R:
Speed is not changing because there is no curvature at that point (ie acceleration is zero)

For point S:
The curve is concave up so a>0 and from the graph the slope of tangent line at S is zero so v=0

Now I have no problem with point P and R. But I am not sure what to say about point Q and S.

Unfortunately, to add to my confusion, the answer in the book says the speed in decreasing for both Q and S.