 Hello and welcome to the session. In this session, first we will discuss increasing function. Here we will use differentiation to find out whether a function is increasing or not. Let i be an open interval contained in the domain of a real valued function then f is said to be increasing on i if x1 less than x2 in i implies f of x1 is less than equal to f of x2 for all x1 x2 belongs to i and also f is said to be strictly increasing on i if x1 less than x2 in i implies f of x1 is less than f of x2 for all x1 x2 belongs to i. Next we will discuss decreasing function. Again the function f is said to be decreasing on i if x1 less than x2 in i implies f of x1 greater than equal to f of x2 for all x1 x2 belongs to i. Also the same function f is said to be strictly decreasing on i if x1 less than x2 in i implies f of x1 is greater than f of x2 for all x1 x2 belongs to i. Consider a function fx equal to 3x plus 5 so let's check if the function fx is strictly increasing or not. For this consider x1 x2 belongs to r such that x1 is less than x2. Now x1 is less than x2 so this implies 3x1 is less than 3x2. Now we add 5 on both the sides so we get 3x1 plus 5 is less than 3x2 plus 5 that is we get f of x1 is less than f of x2 so we get that x1 less than x2 implies f of x1 is less than f of x2 for all x1 x2 belongs to r thus we have fx equal to 3x plus 5 is strictly increasing on r. Next we discuss a very important result which says let f be continuous on closed interval AB and differentiable on open interval AB then f is increasing in closed interval AB if f dash x is greater than 0 for each x belongs to open interval AB also f is decreasing in closed interval AB if f dash x is less than 0 for each x belongs to open interval AB then f is a constant function in closed interval AB if f dash x is equal to 0 for each x belongs to open interval AB also we have that f is strictly increasing in open interval AB if f dash x is greater than 0 for each x belongs to open interval AB f is strictly decreasing in open interval AB if f dash x is less than 0 for each x belongs to open interval AB also a function will be increasing or decreasing in r if it is so in every interval of r. Let's consider the function fx equal to 2x square minus 3x from here we have f dash x equal to 4x minus 3 we need to find the interval in which this function fx is strictly increasing a function f would be strictly increasing if f dash x is greater than 0 so we have f dash x greater than 0 that is 4x minus 3 greater than 0 which implies that x should be greater than 3 by 4 so we get f is strictly increasing in open interval 3 by 4 to infinity now f would be strictly decreasing when f dash x is less than 0 that is we have 4x minus 3 is less than 0 which gives x is less than 3 by 4 so hence we have f is strictly decreasing in open interval minus infinity to 3 by 4 so this is how using the derivatives we can check if a function is increasing or decreasing strictly increasing or strictly decreasing also this completes the session hope you have understood the increasing and decreasing functions