Can the average velocity of a particle moving along the x axis ever beif the acceleration is not constant? Prove your answer with the use of graphs.

No, the average velocity can never be if the acceleration is not constant. It is important to note that the average velocity of between two points in the velocity-time graph lies exactly the halfway between the points only when the those points are located on a straight line i.e. the velocity time graph should be a straight line.
Also if the velocity time graph is a straight line, the acceleration (the slope of the velocity time graph) will always be constant.
For example, consider the velocity time graph of a particle in motion below. The red color plot shows the parabolic relation of velocity of the particle with time while the blue color plot shows the linear relationship of velocity with time.
Plot:


The average velocity between two points on the blue color line will lie exactly at the half way between the two points whereas for the red color plot, the average velocity will not lie exactly at the half way between the points. This is due to the non-linear relationship between the velocity and time in the red color plot.
The non-linearity means that the change in velocity with time is not constant. Therefore the plot of velocity-time (red color) shows a curvature.
The acceleration time graph for the corresponding velocity time is shown below i.e. the red color plot is the acceleration time of the red color velocity time and the blue color plot is the acceleration time of the blue color velocity time.
Plot:


One can easily compare from the two plots above that for the curve whose acceleration is not constant over time, the velocity has non-linear relationships, henceforth the average velocity of the two points will not lie in the half way between the points.