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 )].