 Hi and welcome to the session. I am Deepika here. Let's discuss a question. Find the values of x for which y is equal to x into x minus 2 square is an increasing function. So let's start the solution. Now given y is equal to x into x minus 2 square that is y is equal to x square into x square minus 4x plus 4 and this is equal to x is to power 4 minus 4x cube plus 4x square. This implies dy by dx is equal to 4x cube minus 12x square plus 8x and this is equal to 4x if we take common we get x square minus 3x plus 2 and this is equal to 4x into x minus 1 into x minus 2. Now dy by dx is equal to 0 gives 4x into x minus 1 into x minus 2 is equal to 0 and this implies x is equal to 0, x is equal to 1 or x is equal to 2. Hence the points x is equal to 0, x is equal to 1 and x is equal to 2 divides the real line into 4 disjoint intervals minus infinity to 0 0 to 1 1 to 2 to 2 infinity Now we will check in which interval our dy by dx is greater than 0. In teasing function dy by dx is greater than 0 that is if 4x into x minus 1 into x minus 2 is greater than 0. Now for x less than 0 dy by dx is also less than 0. This implies y is a decreasing function in minus infinity to 0 for 0 less than x x less than 1 dy by dx is greater than 0. This implies y is in an increasing function in an open interval 0 to 1. That is if we take the values of x between 0 and 1 then y is an increasing function. Now for 1 less than x x less than 2 dy by dx is less than 0. This implies y is a decreasing function in an open interval 1 to 2 and again for x greater than 2 our dy by dx is greater than 0. This implies y is an increasing function and open interval 2 to infinity. Friends, the values of x for which y is an increasing function are 0 is less than x, x is less than 1 and x greater than 2 and this is the answer for the above question. I hope the question is clear to you. Bye and take care.