 Hi and welcome to the session. Let us discuss the following question. Question says, find two numbers whose sum is 24 and whose product is as large as possible. First of all, let us understand that if we are given a function f defined on interval i, c is any point belonging to interval i such that f double dash c exists. c is a point of local maxima if f dash c is equal to 0 and f double dash c is less than 0. And c is a point of local minima if f dash c is equal to 0 and f double dash c is greater than 0. This is the key idea to solve the given question. Now let us start the solution. Now we are given two numbers whose sum is 24 and whose product is as large as possible. Now let one number be x, then other number is equal to 24 minus x. Now let product of the numbers be equal to p which is given by x multiplied by 24 minus x. Now we can write product p is equal to 24x minus x square. Now differentiating both sides with respect to x we get dp upon dx equal to 24 minus 2x. Now we are given that the product of the two numbers is as large as possible. So for maximum value of the product we put dp upon dx equal to 0. Now we get 24 minus 2x equal to 0. We know dp upon dx is equal to 24 minus 2x. So we can write 24 minus 2x equal to 0. Now subtracting 24 from both sides we get minus 2x equal to minus 24. Now dividing both sides by minus 2 we get x equal to 12. Now we know dp upon dx is equal to 24 minus 2x. Now differentiating both sides with respect to x we get d square p upon dx square equal to minus 2. Now clearly we can see at x equal to 12 dp upon dx is equal to 0 and d square p upon dx square is equal to minus 2 which is less than 0. So from key idea we get x is equal to 12 is a point of local maxima or we can say product p is maximum at x equal to 12. So we get required two numbers are 12 and 24 minus 12. We know 24 minus 12 is equal to 12 only. So two numbers are 12 and 12. This is our required answer. This completes the session. Hope you understood the session. Keep smiling and have a nice day.