 Hello and welcome to the session I am Deepika here. Let's discuss a question which of the following functions are strictly decreasing on an open interval 0 to pi by 2 a cos x b cos 2x c cos 3x and d tan x. So let's start the solution. Now we know that a function is strictly decreasing on an open interval where its derivative is negative. So for the given question we will check the derivative of which of the following function is negative on an open interval 0 to pi by 2. So let y is equal to cos x therefore dy by dx is equal to minus sin x. Now for 0 less than x less than pi by 2 sin x is greater than 0. This implies minus sin x is less than 0. This implies dy by dx is less than 0. y strictly decreases on an open interval 0 to pi by 2. Let's move to the part b. Now given function is cos 2x. So let y is equal to cos 2x therefore dy by dx is equal to minus 2 sin 2x. Now for 0 less than x less than pi by 2 that is for 0 less than 2x less than pi positive in first and second quadrant therefore dy by dx which is equal to minus 2 sin 2x is negative that is less than 0. y strictly decreases on an open interval 0 to pi by 2. Let's move to the part c. In part c function is cos 3x let y is equal to cos 3x therefore dy by dx is equal to minus 3 sin 3x. Now for 0 less than x less than pi by 2 that is for 0 less than 3x less than 3 pi by 2. Now for 0 less than 3x less than pi sin 3x is positive therefore dy by dx which is equal to minus 3 sin 3x is negative that is less than 0 therefore y strictly decreases on an open interval 0 to pi less than 3x less than 3 pi by 2 sin 3x is negative therefore dy by dx which is equal to minus 3 sin 3x is positive since y strictly increases on an open interval pi to 3 pi by 2 y strictly decreases on an open interval 0 to pi but strictly increases on an open interval pi to 3 pi by 2 therefore y is not a strictly decreasing function on an open interval 0 to pi by 2. Let's move to the part d. Here the given function is tan x let y is equal to tan x therefore dy by dx is equal to sin square x. Now for 0 less than x less than pi by 2 sin square x is positive being a perfect square therefore dy by dx is positive increases on an open interval 0 to pi by 2. Hence the answer for the upper question is that the functions given in part a and part b strictly decreases on an open interval 0 to pi by 2. I hope the solution is clear to you. Bye and check you.