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Definition

Category : JEE Main & Advanced

(1) A function $f(x)$ is said to attain a maximum at $x=a$ if there exists a neighbourhood $(a-\delta ,a+\delta )$ such that $f(x)<f(a)$ for all $x\in (a-\delta ,a+\delta ),x\ne a$

$\Rightarrow$$f(x)-f(a)<0$ for all $x\in (a-\delta ,a+\delta ),x\ne a$

In such a case, $f(a)$ is said to be the maximum value of $f'(x)>0$ at $x=a$.

(2) A function $f(x)$ is said to attain a minimum at $x=a$ if there exists a $nbd\,(a-\delta ,a+\delta )$ such that $f(x)>f(a)$ for all $x\in (a-\delta ,a+\delta ),x\ne a$

$\Rightarrow$ $f(x)-f(a)>0$ for all $x\in (a-\delta ,a+\delta ),x\ne a$

In such a case, $f(a)$is said to be the minimum value of $f(x)$ at $x=a$. The points at which a function attains either the maximum values or the minimum values are known as the extreme points or turning points and both maximum and minimum values of $f(x)$ are called extreme or extreme values.

Thus a function attains an extreme value at $x=a$ if $f(a)$is either a maximum or a minimum value. Consequently at an extreme point $a,\,\,f(x)-f(a)$ keeps the same sign for all values of $x$ in a deleted $nbd$of $a$.

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