 Hi and welcome to the session, let us discuss the following question, question says, at what points in the interval 0 to pi, does the function sin 2x attain its maximum value? First of all, let us understand that if function f is continuous on closed interval kb, then f has absolute maximum value and absolute minimum value, which function f attains at least once in the interval ab. This is the key idea to solve the given question. Now let us start with the solution, let function f given by fx equal to sin 2x. Now differentiating both sides with respect to x, we get f dash x equal to 2 cos 2x. Now to find critical values of x, we put f dash x equal to 0, we get 2 cos 2x equal to 0, now this implies cos 2x equal to 0. Now we know cos pi upon 2 cos 3 pi upon 2 cos 5 pi upon 2 cos 7 pi upon 2 is equal to 0, all these values pi upon 2 3 pi upon 2 5 pi upon 2 7 pi upon 2 lying the closed interval 0, 2 pi. Now cos 2x is equal to 0, this further implies 2x is equal to pi upon 2 or 2x is equal to 3 pi upon 2 or 2x is equal to 5 pi upon 2 or 2x is equal to 7 pi upon 2. Now we get x is equal to pi upon 4 or x is equal to 3 pi upon 4 or x is equal to 5 pi upon 4 or x is equal to 7 pi upon 4. Now we will find the value of fx at x equal to 0 pi upon 4 3 pi upon 4 pi pi upon 4 7 pi upon 4 and 2 pi. Now first of all let us find out f0 which is equal to sin 0 which is simply equal to 0. Now let us find out f pi upon 4. Now this is equal to sin 2 multiplied by pi upon 4 which is equal to sin pi upon 2. We know sin pi upon 2 is equal to 1. So we get f pi upon 4 equal to 1. Now let us find out f 3 pi upon 4. Now this is equal to sin 2 multiplied by 3 pi upon 4 which is equal to sin 3 pi upon 2. Now this is equal to minus 1. Now let us find out f 5 pi upon 4 this is equal to sin 2 multiplied by 5 pi upon 4 which is equal to sin 5 pi upon 2 or we can write it equal to 1. So we get f 5 pi upon 4 equal to 1. Now we will find out f 7 pi upon 4 this is equal to sin 2 multiplied by 7 pi upon 4 or we can write it equal to sin 7 pi upon 2. Now this is equal to minus 1. Now let us find out f 2 pi this is equal to sin 2 multiplied by 2 pi. Now this is equal to sin 4 pi which is equal to 0. Now clearly we can see function attains maximum value at x equal to pi upon 4 and x equal to 5 pi upon 4. So we can write absolute maximum value of function f is 1 that occurs at x equal to pi upon 4 and x equal to 5 pi upon 4. So our required answer is given function attains maximum value at x equal to pi upon 4 and 5 pi upon 4. This completes the session hope you enjoyed the session have a nice day.