1 . To Find Range of a Continuous Function Let f ( x ) be a continuous function on [ a , b ], such that its least value in [ a , b1 is m and the greatest value in [ a , b ] is M . Then , range of value of f ( x ) for x ∈ [ a , b ] is [ m , M ].
2 . To Check for the injectivity of a Function A strictly monotonic function is always one-one ( injective ). Hence , a function f ( x ) is one-one in the interval [ a , b ], if f ‗( x ) > 0 , ∀ x
∈ [ a , b ] or f ‘ ( x ) < 0 , ∀ x ∈ [ a , b ].
3 . The points at which a function attains either the local maximum value or local minimum values are known as the extreme points or turning points and both local maximum and local minimum values are called the extreme values of f ( x ). Thus , a function attains an extreme value at x = a , if f ( a ) is either a local maximum value or a local 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 .
4 . A necessary condition for ( a ) to be an extreme value of a function ( x ) is that f ‗( a ) = 0 in case it exists .
5 . 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 . e . g ., the function f ( x ) = x 3 , f ‗( 0 ) = 0 but at x = 0 the function does not attain an extreme value .
6 . 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 .
7 . All x , for which f ‗( x ) = 0 , do not give us the extreme values . 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 ).
Critical Points of a Function
Points where a function f ( x ) is not differentiable and points where its derivative ( differentiable coefficient ) is z ?, ro are called the critical points of the function f ( x ).