What is the difference between necessary condition & sufficient condition?

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?

• As a side note: Your example of a necessary condition is not a necessary condition - unless you impose that $f$ be differentiable at $a$ Jun 7 '15 at 8:09
• In case you don't understand HagenvonEitzen's comment: consider the absolute-value function $f(x) = |x|$, or e.g. the function $g(x) = e^x$ for $x\neq0$, with $g(0) = 0$. Both have a minimum at $0$, but they aren't actually differentiable there (and for $g$, naïvely trying to differentiate would actually give $g'(0) = 1$!), so $f'(a) = 0$ is not fulfilled. Jun 7 '15 at 14:04
• @leftaroundabout: Yes, my book also cites that but the million dollar question: what should I do to find the extremum point, if $f'(x)$ isn't sufficient like here where the mod function isn't differentiable?
– user142971
Jun 7 '15 at 14:07
• @user36790: for non-differentiable functions, you need to explicitly proove extrema (set up a small interval $I\ni x_{\mathrm{e}}$, and proove that $f(x) < f(x_{\mathrm{e}})$ for all $x \in I\setminus\{x_{\mathrm{e}}\}$). To get an intuitition about finding $x_{\mathrm{e}}$ in the first place, it's helpful to plot the function, often you'll just see where there appears to be a maximum. For instance, the function $x\mapsto x\mod a$ has minima at each multiple of $a$. (Note that these are not maxima, which you don't really see in the plot) Jun 7 '15 at 14:12

$P \Rightarrow Q$ $\quad$ [This is read as "If P, then Q"]

$P$ is a sufficient condition for $Q$
$Q$ is a necessary condition for $P$

• Great answer, but only if you are familiar with $\Rightarrow$ notation. Jun 7 '15 at 19:29
• @PaulDraper Yea you're right! :) I'll do a little editing! Jun 8 '15 at 0:14

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.,

$$\rm Rain \implies Clouds .$$

• This is my favorite example for talking about logical implication. Jun 7 '15 at 11:25
• Useful, but I prefer something like "Oxygen is necessary to make water molecules, but it is not sufficient." Jun 8 '15 at 3:19

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.

Necessary condition $A$ means that $A$ that must exists in order for $B$ to occur, however, $A$ alone doesn't guarantee that $B$ happens.

Sufficient condition $A^\prime$ means that if $A^\prime$ happened then $B$ will inevitably occur.

Getting good grades is a necessary condition to get accepted into a prestigious university

Solving the Riemann hypothesis is sufficient to get accepted into a prestigious university.

So in short, a sufficient condition guarantees the occurrence of another condition

But a necessary condition doesn't guarantee it .

Also assume that $S$ is a sufficient condition for $B$ and $N$ is a necessary condition for $B$.

It is worth mentioning that if $S$ occurred it includes as well that $B$ occurred.

Sufficiency is a stronger notion.

If someone solved Riemann hypothesis,It is very obvious that he have extremely good grades , but it is not true that every one who gets good grades will solve the Riemann hypothesis

The necessary condition is automatically met when the sufficient condition is met.

If the necessary condition is not met, the sufficient condition is automatically not met.

• Typo? "Sufficient condition A′ means that A′ must exists in order for B to occur [...]"; wouldn't that mean that one must solve the Riemann hypothesis to get accepted into a prestigious university? Jun 7 '15 at 8:58
• @Thomas Be careful, Sufficient condition $A^\prime$ doesn't mean it is the only condition that guarantees $B$, Maybe other conditions leads to $B$ as well. However, It means that if it does happen then $B$ will inevitably happen Jun 7 '15 at 9:06
• That's right; but your second sentence contradicts this: you wrote that if A' is a sufficient condition for B then A' must exist in order for B to occur, which is not true. That's why I asked if you had a typo Jun 7 '15 at 9:07
• @Thomas yup I edited it Jun 7 '15 at 9:09

Necessity and sufficiency address the direction of implication of statements.

By stating that $f'(a)=0$ is necessary, your book is claiming that the derivative of $f$ vanishes at every extreme point. (Think of the slope of the tangent at such a point).

By stating that it is not sufficient, your book is referring to examples such as $f(x)=x^3$ and $f(x)=x^5$ at the origin -- these have vanishing derivatives but no extreme points at $a=0$. (Tangent can be horizontal without a maximum or minimum point).

Usually, you need more information such as the sign of the second derivative at the candidate point or the sign of the first derivative around a candidate point to decide if it is an extreme point.