 Hello and welcome to the session. In this session we discussed the following question which says, In a factory, it is found that the number of units x produced in a day depends upon the number of workers n and is obtained by the relation x equal to 7n into nq plus 22. The whole to the power of minus 1 upon 2, the demand function of the product is p equal to 5 upon 3 plus x determine the marginal revenue when m is equal to 3 also show that the marginal revenue decreases with increase of x. If we have that p is the price per unit and x is the number of units demanded then we have the demand function is given as x equal to f of p or p equal to f of x the total revenue R is equal to p into x then the average revenue AR is equal to the total revenue upon the number of units sold that is R upon x then we have the marginal revenue MR is equal to the rate of change of the total revenue with respect to the quantity sold that is dr upon dx is the marginal revenue MR. This is the key idea that we use for this question. Let's now proceed with the solution. In the question we are given that in a factory number of units x are produced in a day and this depends on the number of workers n and the relation between x and n is given by this we are also given the demand function we have to determine the marginal revenue when n is equal to 3 and we also have to show that the marginal revenue decreases with increase of x. Now first of all we have the demand function given as p equal to 5 upon 3 plus x as we know that the total revenue capital R is equal to p into x so total revenue capital R is equal to p that is 5 upon 3 plus x the whole into x and from here we have R is equal to 5x upon 3 plus x this is the total revenue. Now marginal revenue is dr upon dx so let's now find out the marginal revenue MR which is equal to dr by dx that is d by dx of 5x upon 3 plus x so this is equal to in the denominator we have 3 plus x the whole square minus the numerator we have 3 plus x the whole into d by dx of 5x which is 5 minus 5x into d by dx of 3 plus x which is 1 so this is equal to 15 plus 5x minus 5x the whole upon 3 plus x the whole square 5x minus 5x cancels and we have the marginal revenue MR is equal to 15 upon 3 plus x the whole square let this be 1 now the relation between x and n is given as x equal to 7n into n cube plus 22 we hold to the power of minus 1 upon 2 now putting n equal to 3 in this we get x is equal to 7 into 3 the whole into 3 cube plus 22 the whole to the power of minus 1 upon 2 that is we have x is equal to 21 into 27 plus 22 we hold to the power of minus 1 upon 2 or further we have x equal to 21 upon square root of 49 which is equal to 21 upon 7 now 7 3 times is 21 so this gives us x equal to 3 thus we can say for n equal to 3 we get x also as 3 as we are supposed to find out the marginal value when n is equal to 3 so for this we will put x as 3 in relation 1 since x is equal to 3 when n is equal to 3 so now putting x equal to 3 in 1 we have marginal revenue MR is equal to 15 upon 3 plus 3 the whole square that is 15 upon 6 whole square which is 36 now 3 5 times is 15 and 3 12 times is 36 so we have the marginal revenue when n is equal to 3 is 5 upon 12 so we have got this value now next we are supposed to show that the marginal revenue decreases with increase in x for this we now find out d by dx of the marginal revenue MR now as MR is equal to 15 upon 3 plus x the whole square so now d by dx of MR would be equal to 15 into d by dx of 1 upon 3 plus x the whole square or we have d by dx of the marginal revenue MR is equal to 15 into minus 2 upon 3 plus x whole cube so this is equal to minus 30 upon 3 plus x the whole cube is d by dx of the marginal revenue MR now x will take only positive values so we can say for all values of x 3 plus x whole cube is positive only positive values as x denotes the number of units so as for all values of x this denominator would be positive so we can say that d by dx of the marginal revenue MR would be less than 0 as this is less than 0 hence we can say that the marginal revenue decreases with increase of x this is what we were supposed to show so this completes the session hope you have understood the solution of this question