 Hello students, let's work out the following problem. It says if the cost function of article manufactured by accompanying is given by Cx is equal to 300x minus 10x square plus 1 by 3x cube, find the output at which marginal cost is minimum. The average cost is minimum. So let's now move on to the solution. The cost function Cx is given as 300x minus 10x square plus 1 by 3x cube. Now, marginal cost is mc is d by dx of C. So this is d by dx of C is 300 minus 20x plus 3 by 3x square. So this is 300 minus 20x plus x square. Now, to minimize the marginal cost, we need to find the first order derivative of the marginal cost. So d by dx of mc is minus 20 plus 2x. Now for minimum or for marginal cost mc to be minimum d by dx of mc that is minus 20 plus 2x is equal to 0. Now we have minus 20 plus 2x is equal to 0. So this implies 2x is equal to 20 and this implies x is equal to 10. Now we need to find the second order derivative of d by dx of mc. So this is 2 that is derivative of d by dx of mc is 2 which is greater than 0. So this implies marginal cost is minimum when x is equal to n. Now in the second part we have to minimize the average cost. The cost function is given as 300x minus 10x square plus x cube by 3. Now the average cost that is ac is given by c by x that is we divide this by x. So this is equal to 300 minus 10x plus x square by 3. Now again for minimizing we need to find the first and the second order derivative of the average cost. Now d by dx of ac is minus 10 plus 2 by 3x. Now we put it equal to 0 for average cost to be minimum d by dx of ac that is minus 10 plus 2 by 3x is equal to 0. So this implies 2 by 3x is equal to 10 and this implies 2x is equal to 30 and this implies x is equal to 30 by 2 that is 15. Now we find the second order derivative of average cost. Now this is 2 by 3x the derivative would be 2 by 3 which is greater than 0. So this implies average cost is minimum when x is equal to 15. So the marginal cost is minimum when x is equal to 10 that is the output is 10 units and average cost is minimum when x is equal to 15 that is output is 15 units. So this completes the question and the session. Bye for now. Take care. Have a good day.