 Hi and welcome to the session. Let us discuss the following question. Question says, find the maximum and minimum values if any of the following functions given by fx equal to modulus of sin 4x plus 3. First of all, let us understand that if function f is defined or interval i, then if there exists c belonging to interval i such that fc is greater than equal to fx, then fc is called the maximum value of function f in interval i. If fc is less than equal to fx, then fc is called the minimum value of function f in interval i. This is the key idea to solve the given question. Let us now start the solution. We know function f is given by fx equal to modulus of sin 4x plus 3. Now we know that sin function has maximum value 1 and minimum value minus 1 in real numbers. So we can write sin 4x is less than equal to 1 and greater than equal to minus 1. Now this implies minus 1 plus 3 is less than equal to sin 4x plus 3 and sin 4x plus 3 is less than equal to 1 plus 3. This implies sin 4x plus 3 is greater than equal to 2 and less than equal to 4. This implies modulus of sin 4x plus 3 is less than equal to modulus of 4 and greater than equal to modulus of 2. This implies you know modulus of 2 is equal to 2 and modulus of 4 is equal to 4. So we can write modulus of sin 4x plus 3 is less than equal to 4 and greater than equal to 2. This implies you know modulus of sin 4x plus 3 is equal to fx. So we can write fx for it. Now we get fx is greater than equal to 2 and less than equal to 4. Now 2 is less than equal to fx implies 2 is the minimum value of function f. So we can write minimum value of function f is 2 and fx is less than equal to 4 implies 4 is the maximum value of function f. So this is our required answer. This completes our session. Hope you understood the session. Take care and good bye.