JEE Main & Advanced Mathematics Applications of Derivatives Definition

(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 \[a\].