The slope of a position-time graph reveals informaiton about the speed of an object. In fact one could generalize that "however the slope goes, so goes the velocity." A constant velocity motion is represented by a line of constant slope and a changing velocity by a line of changing slope. A positive velocity motion is represented by a line of positive slope and a negative velocity by a line of negative slope. A small velocity motion (slow object) is represented by a line with a small slope and a large velocity by a line with a large slope. In the end, by observing the slope, one can make conclusions about the velocity.
 

In this question, you must identify the one stage where the speed is smallest (for moving slowest) or largest (for moving fastest).  The speed is the "magnitude" of the velocity. It can be thought of as the velocity without a direction (or without a positive or a negative sign). This is to say, an object with a velocity of -20 m/s is moving faster than an object with a velocity of 10 m/s.

Applying this principle to the graph means that you are looking for the one stage during which the line slopes up or down the most steeply (for fastest-moving object) or slopes up or down the most gently (for slowest-moving object). You do not want to pick any of the horizontal lines because the object is not moving during this time. You are looking for a moving object with the greatest or the smallest speed.

So read the question carefully to see what motion characteristic you are looking for - either the greatest or the smallest speed. Then inspect each sloped section to identify the section with the greatest slope (fastest speed) or the smallest slope (moving with the smallest speed).