f( x) is said to be non-increasing in D1, if for every x1, x2 ∈ D1, x1 > x2 ⇒ f( x1) ≤ f( x2). It means that the value of f( x) would never increase with an increase in the value of x.
If a function is either strictly increasing or strictly decreasing, then it is also a monotonic function.
Important Points to be Remembered
( i) A function f( x) is said to be increasing( decreasing) at point x0, if there is an interval( x0— h, x0 + h) containing x0, such that f( x) is increasing( decreasing) on( x0— h, x0 + h).
( ii) A function f( x) is said to be increasing on [ a, b ], if it is increasing( decreasing) on( a, b) and it is also increasing at x = a and x = b.
( iii) If( x) is increasing function on( a, b), then tangent at every point on the curve y = f( x) makes an acute angle θ with the positive direction of x-axis.
∴
( iv) Let f be a differentiable real function defined on an open interval( a, b).
If f ‗( x) > 0 for all x ∈( a, b), then f( x) is increasing on( a, b). If f ‗( x) < 0 for all x ∈( a, b), then f( x) is decreasing on( a, b).
( v) Let f be a function defined on( a, b).
If f ‗( x) > 0 for all x ∈( a, b) except for a finite number of points, where f ‗( x) = 0, then f( x) is increasing on( a, b).
If f ‗( x) < 0 for all x ∈( a, b) except for a finite number of points, where f ‗( x) = 0, then f( x) is decreasing on( a, b).
Properties of Monotonic Functions
1. If f( x) is strictly increasing function on an interval [ a, b ], then f-1 exist and also a strictly increasing function.
2. If f( x) is strictly increasing function on [ a, b ], such that it is continuous, then f-1 is continuous on [ f( a), f( b)].