Monotonocity:
Let y = f (x) be a given function with 'D' as it is domain. Let D1 ⊆ D2 then;
Increasing Function:
If a function f(x) is satisfying x1 > x2 => f(x1) > f (x2) for all x1, x2 ∈ D1, it means that value of f (x) will keep on increasing with the increase in value of x, then f is called as increasing in D1.
Decreasing Function:
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If a function f(x) is satisfying x1 > x2 => f(x1) < f (x2) for all x1, x2 ∈ D1, it means that the value of f (x) will keep on the increasing with an increase in value of x, then f is called as decreasing in D1.
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Non-Decreasing Function:
If a function f(x) is satisfying x1 > x2 => f(x1) ≥ f (x2) for all x1, x2 ∈ D1, it means that the value of f (x) will never decrease with an increase in the value of x, then f is called non-decreasing in D1.
Non-Increasing Function:
If the function f(x) is satisfying x1 > x2 => f(x1) ≤ f (x2) for all the x1, x2 ∈ D1, it means that value of f (x) will never increase with the increase in value of x, then f is called as non-increasing in D1.
Note:
(i) If 
and points which make f'(x) equal to zero (in between (a, b)) do not form an interval, then f (x) would be increasing in [a, b] else it will be non-decreasing function.
(ii) I f and points which make f'(x) equal to zero (in between (a, b)) do not form an interval, f (x) would be decreasing in [a, b], else it will be non increasing.
Monotonic Function:
A function which is increasing or decreasing in its domain is called as monotonic function.
Illustration: Show that function f(x) = 2 cos x + cot x + 3x is decreasing in (0,Π))
Solution:
2sin x + 1 > 0 in (0, Π) hence f' > 0 function is increasing.
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