 Hello and welcome to the session, let's discuss the following question. It says, find the interval in which the function f given by fx is equal to sin x minus cos x, x lying between 0 to 2 pi is increasing or decreasing. So let's now move on to the solution. The given function is sin x minus cos x. Now to find the interval in which the function is increasing or decreasing, we need to find the derivative of the function that is f dash x. Our derivative of sin x is cos x and the derivative of cos x is minus sin x. So this becomes plus sin x. Now this can be again written as root 2 into cos x into sin pi by 4 because we know that sin pi by 4 is 1 by root 2 plus sin x cos pi by 4. Again cos pi by 4 is 1 by root 2. Now this is equal to root 2 into sin x plus pi by 4. This is by the formula sin a plus b is equal to sin a plus sin a cos b plus sin b cos a. Now for the function to be increasing, we must have f dash x to be greater than 0. So this implies root 2 into sin x plus pi by 4 is greater than 0. So this implies sin x plus pi by 4 is greater than 0. Now since sin x plus pi by 4 is greater than 0, that means x plus pi by 4 lies between 0 to pi. As we know that if sin theta is greater than 0, then theta lies between 0 to pi. And here theta is x plus pi by 4. So we have 0 is less than x plus pi by 4 is less than pi. So we have minus pi by 4 is less than x is less than pi minus pi by 4 is 3 by 4. So this can be again written as minus pi by 4 is less than x is less than 0 or 0 is less than x is less than 3 pi by 4. Now the given interval is 0 to 2 pi. And we see that these values of x that is when x lies between minus pi by 4 to 0 lies outside the interval 0 to 2 pi. So we ignore this interval and we have x belonging to the open interval 0 to 3 pi by 4. So this implies fx is increasing on the interval 0 to 3 pi by 4. Now for fx to be decreasing we must have f dash x less than 0 that is root 2 into sin x plus pi by 4 is less than 0. So this implies sin x plus pi by 4 is less than 0. Now sin theta is less than 0 that means theta lies between pi to 2 pi. That means x plus pi by 4 lies between pi to 2 pi. So this implies pi minus pi by 4 is less than x is less than 2 pi minus pi by 4. So this implies 3 pi by 4 is less than x is less than 7 pi by 4. Hence fx decreasing on the interval 3 pi by 4 to 7 pi by 4. So the given function is increasing on the interval 0 to 3 pi by 4 and it is decreasing on the interval 3 pi by 4 to 7 pi by 4. So this completes the question and the session. Bye for now. Take care. Have a good day.