Increasing Function: A function f(x) is said to be an increasing function on (a,b) if 

x₁ ≤ x₂ ⇒ f(x₁ ) ≤ f(x₂ ) for all x₁ , x₂ ∈ (a,b)

Example: Show that the function f(x) = cot^-1 x + x is increasing in the interval (-∞,∞)

Solution:

Given f(x) = cot^-1 x + x

Differentiate it with respect to x, we get

       g(x) = -1/(1+ x² ) + 1                              {let diffrenetitaion on f(x) is g(x)}

=>  g(x) = (-1 + 1 + x² ) /(1+ x² )

=> g(x) = x² /(1+ x² )

For a function to be increasing, its derivative must be greater than zero

i.e  g(x) > 0

Now g(x) = x² /(1+ x² ) > 0

Here g(x) is greter than zero for all values of x

So f(x) is increasing on (-∞,∞)

Decreasing Function: A function f(x) is said to be a decreasing function on (a,b) if 

x₁ ≤ x₂ ⇒ f(x₁ ) ≥ f(x₂ ) for all x₁ , x₂ ∈ (a,b)

Example: Show that the function f(x) = x^x is decreasing on the interval (0,1/e)

Solution:

Given f(x) = x^x 

Differentiate it with respect to x, we get

       g(x) = x^x *(1 + logx)                                                                  {let diffrenetitaion on f(x) is g(x)}

For a function to be decreasing, its derivative must be less than zero

i.e  g(x) < 0

Now g(x) = x^x *(1 + logx) < 0

=> x^x *(1 + logx) < 0

=> 1 + logx < 0

=> logx < -1

=> x < e^-1

=> x < 1/e

So f(x) is decreasing on (0,1/e)

In this way we can know that a particular function is increasing or decreasing at a particular interval.