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

## Necessary Condition for Extreme Values

Category : JEE Main & Advanced

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.

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