TITLE: What is the difference between necessary condition & sufficient condition?
QUESTION [12 upvotes]: My book says :

For having extreme point $a$ of function $f$, the necessary condition is that $f'(a) = 0$. However, it isn't a sufficient condition. 

Now, what is the difference between necessary & sufficient condition? And also what is the sufficient condition for a function to have an extreme point?

REPLY [15 votes]: If it is raining, it is $\textrm{sufficient}$ to conclude that there are clouds. However, presence of clouds is not enough to conclude that there will be rain, but clouds are $\textrm{necessary}$ to have rain i.e.,
$$\rm Rain \implies Clouds .$$

REPLY [5 votes]: Look at this example:

Every time it is raining, the street gets wet.  

But there are also other possibilities to wetten the street. For example: On a bright day without any clouds in the sky someone could pour water on the street.
sufficient
If you look out of the window, and see that it is raining, without seeing the street, you know for sure, that the street is wet. So the condition "It is raining« is sufficient to conclude, that the street is wet. You don't need to know anything more.
necessary
On the other side: Your partner comes home with wet shoes, and when asking why the shoes are wet, you get the answer: »The street is wet.« (Your partner never lies to you) Do you now know if it's raining or not?
No, you don't. Someone might have poured water on the street, and you partner might have walked through this puddle on a bright cloudless day. But it is also possible that it is raining.
When you want to proof that it is raining, and you get notice of a dry street, then you know that your proof will fail, because it can't be raining when the street remains dry. So the condition »The street is wet« is a necessary condition.