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)}