Let us suppose that there is a function f(x). There may be some points in the domain where the function does not attain the greatest or least values but the values at these points are greater than or less that the values of the function at the neighbouring points. These points are called as the points of local minimum or local maximum.

Local Maxima: A function f(x) is said to attain a local maximum at x  = a if there exist a neighbour (a-ξ, a+ξ) such that

f(x)<f(a) for all x ∈ (a-ξ, a+ξ) and x is not equal to a

or f(x)- f(a) < 0 for all x ∈ (a-ξ, a+ξ) and x is not equal to a

In this case f(a) is called the local maximum value of f(x) at x = a.

Local Minima: A function f(x) is said to attain a local maximum at x  = a if there exist a neighbour (a-ξ, a+ξ) such that

f(x)>f(a) for all x ∈ (a-ξ, a+ξ) and x is not equal to a

or f(x)- f(a) > 0 for all x ∈ (a-ξ, a+ξ) and x is not equal to a

In this case f(a) is called the local minimum value of f(x) at x = a.