Which graph represents the higher derivatives of a car's position function?

[dekoi]

This question is more theory than anything else.

I am given the graph attached:

This graph shows four functions.
1.) Position function of a car. ($$f(x)$$)
2.) Velocity of the car. ($$f'(x)$$)
3.) Acceleration of the car. ($$f''(x)$$)
4.) Jerk of the car. ($$f'''(x)$$)

I have to identify which graph is which (out of a, b, c, & d), and explain why this is so.

However, I have no idea how to do this.
I am assuming that the order is A, B, C, then D. However, this is based solely on observing the shapes of the graphs (since higher derivatives become more linear), but I do not know the theory to explain this.

Any input is greatly appreciated. Thank you.

 

[TD]

I can't see the attachment yet, but when a certain function reaches a (local) extreme value (minimum or maximum), its derivative is zero. This is also true when the initial function has a horizontal tangent line, which is more general than having an extreme value (it allows e.g. inflections points too).

 

[dekoi]

Can somebody please approve this attachment?
This is an urgent question.

 

[Jameson]

Some basic advice to problems like this is to look at the shapes of the graphs. Do they look like piecewise functions or polynomials? If they look like polynomials, then you know that with each derivative the graph will lose a power.