JEE Main & Advanced Mathematics Applications of Derivatives Necessary Condition for Extreme Values

 A necessary condition for \[f(a)\]to be an extreme value of a function \[f(x)\]is that \[f'(a)=0\], in case it exists.

Note : (1) This result states that if the derivative exists, it must be zero at the extreme points. A function may however attain an extreme value at a point without being derivable there at.

For example, the function \[f(x)=|x|\] attains the minimum value at the origin even though it is not differentiable at \[x=0\].

(2) This condition is only a necessary condition for the point \[x=a\] to be an extreme point. It is not sufficient i.e., \[f'(a)=0\] does not necessarily imply that \[x=a\] is an extreme point. There are functions for which the derivatives vanish at a point but do not have an extreme value there at e.g. \[f(x)={{x}^{3}}\]at \[x=0\]does not attain an extreme value at \[x=0\] and \[f'(0)=0\].

(3) Geometrically, the above condition means that the tangent to the curve \[y=f(x)\] at a point where the ordinate is maximum or minimum is parallel to the x-axis.

(4) The values of \[x\] for which \[f'(x)=0\] are called stationary values or critical values of \[x\] and the corresponding values of \[f(x)\] are called stationary or turning values of \[f(x)\].

(5) The points where a function attains a maximum (or minimum)  are also known as points of local maximum (or local minimum) and the corresponding values of \[f(x)\] are called local maximum (or local minimum) values.