. . Monotonicity of functions Monotonicity of functions are its behaviour of increasing or decreasing. Definition .
A function f x ( ) is said to be an increasing function in an interval I if a f a f b a b I ⇒ ∀ ( ), Definition . A function f x ( ) is said to be a decreasing function in an interval I if a f a f b a b I ⇒ ≥ ∀ ( ), The function f x ( ) = is an increasing function in the entire real line, whereas the function ( ) = − is a decreasing function in the entire real line. In general, a given function may be increasing in some interval and decreasing in some other interval, say for instance, the function f x | | is decreasing in ( , ] −∞ and is increasing in [ , ¥ . These functions are simple to observe for their monotonicity.
But given an arbitrary function how we determine its monotonicity in an interval of a real line? That is where following theorem (stated without proof) will be useful. Theorem . If the function f x ( ) is differentiable in an open interval ( , ) a b then we say, ( ) if d dx f x ( ( )) ≥ , ∀∈ a b , ...
( ) then f x ( ) is increasing in the interval ( , ) a b , ( ) if d dx f x ( ( )) > , ∀∈ a b , ... ( ) then f x ( ) is strictly increasing in the interval ( , ) a b . The proof of the above can be observed from Theorem . .
( ) f x ( ) is decreasing in the interval ( , ) a b if - - Applications of Differential Calculus d dx f x ( ( )) £ , ∀∈ a b . ...( ) ( ) f x ( ) is strictly decreasing in the interval ( , ) a b