 Hello and welcome to the session. In this session we discuss the following question that says, find the values of x for which fx equal to x into x minus 2, the whole square is an increasing function. Also, find the points on the curve where the tangent is parallel to x axis. Before we move on to the solution, let's recall some results. First, we have if a function f be continuous on the closed interval a, b, defensible the open interval a, b, then the function f is increasing in the closed interval a, b. We have greater than 0 for every the open interval a, b. Then it means that the slope of the idea that we use in this question. Let's now proceed with the solution. We are given a function fx and we have to find the values of x for which this function is an increasing function. We have fx is equal to x into this whole square minus fx is equal to x square minus 2 in both sides. We get f dash x is equal to x square plus 8x this means that f dash x is equal to plus 2 the whole is equal to 4x into the factorizing this polynomial. In the many terms we get equal to 4x into this 1 the whole f minus 1 the whole or you can say equal to 4x into x minus 2 the whole into x minus 1 the whole. This means that 4x into f minus 2 the whole into f minus 1 the whole is equal to 0. Which gives us x equal to 0 equals 0 1 and 2 plus infinity 0. Then open into 1 0 1 open into 1 1 2 4 intervals and then we will find the nature of the function fx whether it is increasing or decreasing in the given intervals. To draw the open interval minus infinity 0 is the value of x would be less than 0 is equal to 4x into x minus 1 the whole into negative. Then x minus 1 would be negative and x minus 2 would also be f dash x would be negative or f dash x would be less than 0. Now we know that the function f is increasing if f dash x is greater than 0. So in the same way if f dash x is less than 0 the function f would be decreasing. So we can say that the function fx the open interval minus infinity 0. Now consider the next open interval 0 1 that is the value of x lies between 0 and 1. This would be 4 into positive that is x would be positive and x minus 1 would be negative since the value of x would be less than 1. 2 would also be this would be less than b greater than 0 that is it would be positive. So the function fx would be increasing function here the value of x lies between 1 and 2 dash x. So this would be 4 into for x would be positive then x minus 1 would be 2 would be negative as the value of x would be less than 0 that is it would be negative. Function fx is a decreasing function for the open interval 2 infinity this would be 4 into x would be positive. Now in this case would also be positive since the value of x would be greater than 2 then x minus 2 would also be positive as the value of x would be greater than x would be greater than 0 that is it is positive and hence the function fx would be increasing. So this is between 0 and 1 as the required values of the curve where the tangent is parallel. Now we already know that if a tangent at the slope of the tangent is 0 x equal to x into x minus 2 the whole square x is equal to minus 1 the whole into is equal to 0 this means for x into x minus 1 the whole into x minus 2 the whole is equal to 0 which gives us x equal to 0 though we have f of 0 would be equal to that is we put 0 in place of x in the function fx that is this function. Now we get 0 into 0 minus 2 whole square which would be 0 then we find f of 1 this would be equal to 1 into 1 minus 2 the whole square which would be equal to x equal to 2 we find f of 2 and this would be equal to 2 into the whole square which would be equal to 0. Yes we have found out the points at the tangent our final answer should hope you have understood the solution of this question.