5 . A minimum value at some point may even be greater than a maximum values at some other point .
Maximum and Minimum Values in a Closed Interval
Let y = f ( x ) be a function defined on [ a , b ]. By a local maximum ( or local minimum ) value of a function at a point c ∈ [ a , b ] we mean the greatest ( or the least ) value in the immediate neighbourhood of x = c . It does not mean the greatest or absolute maximum ( or the least
or absolute minimum ) of f ( x ) in the interval [ a , b ]. A function may have a number of local maxima or local minima in a given interval and even a local minimum may be greater than a relative maximum .
Local Maximum
A function f ( x ) is said to attain a local maximum at x = a , if there exists a neighbourhood ( a – δ , a + δ ), of c such that , f ( x ) < f ( a ), ∀ x ∈ ( a – δ , α + δ ), x ≠ a or f ( x ) – f ( a ) < 0 , ∀ x ∈ ( a – δ , α + δ ), x ≠ a
In such a case f ( a ) is called to attain a local maximum value of f ( x ) at x = a .
Local Minimum f ( x ) > f ( a ), ∀ x ∈ ( a – δ , α + δ ), x ≠ a or f ( x ) – f ( a ) > 0 , ∀ x ∈ ( a – δ , α + δ ), x ≠ a
In such a case f ( a ) is called the local minimum value of f ( x ) at x = a . Methods to Find Local Extremum
1 . First Derivative Test Let f ( x ) be a differentiable function on an interval I and a ∈ I . Then ,
1 . ( i ) Point a is a local maximum of f ( x ), if ( a ) f ‗( a ) = 0
( b ) f ‗( x ) > 0 , if x ∈ ( a – h , a ) and f ‘ ( x ) < 0 , if x ∈ ( a , a + h ), where h is a small but positive quantity .
2 . ( ii ) Point a is a local minimum of f ( x ), if