Maximum and minimum values of a function f ( x ) can occur only at critical points . However , this does not mean that the function will have maximum or minimum values at all critical points . Thus , the points where maximum or minimum value occurs are necessarily critical Points but a function may or may not have maximum or minimum value at a critical point .
Point of Inflection
Consider function f ( x ) = x 3 . At x = 0 , f ‗( x )= 0 . Also , f ―( x ) = 0 at x = 0 . Such point is called point of inflection , where 2nd derivative is zero . Consider another function f ( x ) = sin x , f ―( x )= – sin x . Now , f ―( x )= 0 when x = nπ , then this points are called point of inflection .
At point of inflection 1 . It is not necessary that 1st derivative is zero .
2 . 2 nd derivative must be zero or 2nd derivative changes sign in the neighbourhood of point of inflection .
Concept of Global Maximum / Minimum Let y = f ( x ) be a given function with domain D .
Let [ a , b ] ⊆ D , then global maximum / minimum of f ( x ) in [ a , b ] is basically the greatest / least value of f ( x ) in [ a , b ].
Global maxima / minima in [ a , b ] would always occur at critical points of f ( x ) with in [ a , b ] or at end points of the interval .
Global Maximum / Minimum in [ a , b ]
In order to find the global maximum and minimum of f ( x ) in [ a , b ], find out all critical points of f ( x ) in [ a , b ] ( i . e ., all points at which f ‗( x )= 0 ) and let f ( c1 ), f ( c2 ) ,…, f ( n ) be the values of the function at these points .
Then , M1 → Global maxima or greatest value . and M1 → Global minima or least value .
where M1 = max { f ( a ), f ( c1 ), f ( c1 ) ,…, f ( cn ), f ( b )} and M1 = min { f ( a ), f ( c 1 ), f ( c 2 ) ,…, f ( c n ), f ( b )}