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