 Hello and welcome to the session. Let's discuss the following question It says the cost function of a firm is given by see x is equal to 300 x minus 10 x square plus 1 by 3 x cube where x stands for the output Calculate the output at which marginal cost is minimum the output at which average cost is equal to the marginal cost So let's now move on to the solution the cost function x is given by 300 x minus 10 x square plus 1 by 3 x cube Now marginal cost in C is given by d by dx of Cx so mc is equal to d by dx of 300 x minus 10 x square plus 1 by 3 x cube Which is equal to 300 minus 20 x plus x square now To maximize or minimize we need to differentiate this so d by dx of mc is equal to minus 20 plus 2 x now d by dx of mc put it equal to 0 So that gives us minus 20 is Equal to minus 2 x so this implies x is equal to 10 now we see that the second order derivative of the marginal cost will be greater than 0 at x is equal to 10 so d square by dx square of mc is minus 20 when x is equal to 10 Since the first order derivative is minus 20 plus 2 x so derivative of constant is 0 and derivative of 2 x is 2 Which is greater than 0 so this implies the marginal cost is Minimum is equal to 10 so this is the answer to the first part It's now move on to the second part now We have to find the value of the output x for which marginal cost is equal to the average cost Now cx the cost function is given by 300 x minus 10 x square plus 1 by 3 x cube now the average cost is given by Cx upon x So average cost a c is equal to 300 x minus 10 x square plus 1 by 3 x cube Upon x so this is equal to 300 minus 10 x plus 1 by 3 x square now we have to find the value of x for which marginal cost is equal to the average cost Right, so marginal cost is 300 minus 20 x plus x square is equal to the average cost which is 300 minus 10 x plus 1 by 3 x square 300 gets cancelled on both sides. We have minus 10 x plus 1 by 3 x square plus 20 x minus x square is equal to 0 So this implies 10 x plus 1 by 3 x square minus x square is equal to 0 Now taking LCM We have 30 x plus x square minus 3 x square is equal to 0 Now taking x common we have 30 plus x minus 3 x is equal to 0 So we have 30 plus minus 2 x is equal to 0 x minus 3 x is minus 2 x so this implies 2 x is equal to 30 and this implies x is equal to 15 hence average cost That is ac is equal to mc when x is equal to 15 units. So this completes the second part also and the question. Bye for now. Take care. Have a good day