 Hello and welcome to the session. In this session we discussed the following question which says the relation between total cost c and the output x is given by c is equal to 4x into x plus 9 upon x plus 7 the whole plus 3 proves that the marginal cost falls continuously as the output increases. Before we move on to the solution let's discuss what is the marginal cost that is mc this is equal to dc by dx where the c is the total cost and x is the number of itals produced and the average cost that is ac is equal to c upon x that is total cost upon the number of items produced. Now we have a relation between ac and mc which is d by dx of ac is equal to 1 upon x into the marginal cost minus the average cost the whole. Now if we have the marginal cost is greater than the average cost then in that case d by dx of the average cost would be greater than 0 and this means that ac increases with x and the ac curve is rising then next if we have the marginal cost is equal to the average cost then d by dx of ac is equal to 0 which means that ac is constant and the average cost curve remains constant at all levels. Then next condition that we have is if the marginal cost is less than the average cost then we have d by dx of ac is less than 0 which means that the average cost decreases with x ac curve is falling this is the key idea that we use for this question. Now we proceed with the solution in the question we are given the relation between the total cost which is c and the output that is x as this and we have to prove that the marginal cost falls continuously as the output increases. We know that the marginal cost is dc by dx so now we have the function c which is the total cost function that is cx is equal to 4x into x plus 9 upon x plus 7 the whole plus 3. Now the marginal cost which is mc is equal to dc by dx so mc is equal to d by dx of c which is 4x into x plus 9 upon x plus 7 the whole plus 3 the whole. Or we can write this as mc is equal to d by dx of 4x square plus 36x upon x plus 7 plus 3 the whole. So this gives us marginal cost mc is equal to x plus 7 the whole square of the denominator. Then in the numerator we have x plus 7 the whole into d by dx of this which is 8x plus 36 the whole minus 4x square plus 36x the whole into 1. Now this whole plus d by dx of 3 which is 0 so further we get mc is equal to 8x square plus 36x plus 56x plus 252 minus 4x square minus 36x and this whole upon plus 7 the whole square. Further mc is equal to 4x square plus 56x plus 252 upon x plus 7 the whole square. Then mc is equal to taking 4 commons on the numerator we have 4 into x square plus 14x plus 63 upon x plus 7 the whole square the whole. Now x square plus 14x plus 63 can be written as x square plus 14x plus 49 plus 63 minus 49 so further we can write x square plus 14x plus 49 as x plus 7 the whole square plus 63 minus 49 which is 14. So here we have marginal cost mc is equal to 4 into x plus 7 the whole square plus 14 upon x plus 7 the whole square the whole or we can write this as mc is equal to 4 into 1 plus 14 upon x plus 7 the whole square the whole. Now as we had the condition that to show that ac decreases with x we have to show that dy dx of ac is less than 0 and this will prove that ac curve is falling. Similarly as we are supposed to prove that the marginal cost falls continuously as the output increases we need to show that dy dx of mc is less than 0. So next we will find out dy dx of the marginal cost mc that is dy dx of 4 into 1 plus 14 upon x plus 7 the whole square so this is equal to 4 into dy dx of 1 plus 14 upon x plus 7 the whole square the whole. Which is equal to 4 into dy dx of 1 is 0 plus 14 into minus 2 upon x plus 7 the whole square that is we get dy dx of the marginal cost mc is equal to 4 into 14 into minus 2 upon. x plus 7 the whole cube which is equal to minus 112 upon x plus 7 the whole cube this is dy dx of the marginal cost mc now this is less than 0 for all values of x as we know that x would be always greater than 0. We can say that as dy dx of the marginal cost is less than 0 therefore the marginal cost mc falls continuously as the output increases hence we have proved this so this completes the session hope you understood the solution of this question.