 Hello and how are you all today? The question says, the manufacturing cost of an item consists of Rs 900 as overheads. The material cost is Rs 3 per item and the labour cost is Rs x square upon 100 for x items produced. How many items must be produced to have average cost minimum? So here, we are given the parts of our total cost function. First of all, let the number of items produced be equal to x. Right? We are given overheads as Rs 900, material cost as Rs 3 per item and labour cost as Rs x square upon 100 for x items. So this means the total cost function will be equal to our labour cost that is for x items already given to us plus Rs 3 per item means 3 into x because number of items we are producing are x plus the overhead expenses that is 900. So first of all this is our total cost function. Further, with the help of this total cost function we will find out the average cost function. That is equal to the total cost function divided by x. So we can write it like this also. So we have x upon 100 plus 3 plus 900 upon x as our average cost function. Now we know that average cost will be minimum when the first derivative of the average cost function is equal to 0 and the second derivative of the average cost function let it be AC will be greater than 0. So let us first find out the first derivative of the average cost function. We have dy dx of the average cost function as dy dx of x upon 100 plus 3 plus 900 upon x that gives us the answer as x upon sorry it is 1 upon 100 minus 900 upon x square. So this is the first derivative of the average cost function. Now let us equate the first derivative of the average cost function to 0 to obtain the value of x. So we have 1 by 100 minus 900 upon x square equal to 0. This implies 900 upon x square is equal to 1 upon 100. Further implies x square is equal to 900 into 100 that gives us the value of x as plus minus 300 since an output cannot be in negative. So we can see that the value of x is equal to 100 that is negative sign is neglected. Now let us find out the second derivative of the average cost function will be dy dx of dy dx of the average cost function that we have obtained above. So we have dy dx of 1 upon 100 minus 900 upon x square that is further equal to 900 upon x cube and we can see that 900 upon x cube will also will always be greater than 0. So therefore average cost function is minimum 300 items are produced. Right so this completes the session hope you understood it well and enjoyed it also. Bye for now.