 Hello and how are you all today? The question says if the total cost function is given by c is equal to a plus bx plus cx square where x is the quantity of output show that d by dx of the average cost function is equal to 1 by x, marginal cost function minus average cost function. So first of all with the help of the given cost function we will be looking out the average cost function and the marginal cost function. Cost function is given to us as a plus bx plus cx square. Average cost function will be dividing the cost function by x that is a by x plus b plus cx and marginal cost function will be finding out the derivative of the cost function that is b plus 2 cx. Now in the question we need to show this so we will be solving the left hand side separately and the right hand side separately. Let us divide our pay into two column. The left hand side we need to find out the value of that is the derivative of the average cost function that is d by dx of a upon x plus b plus cx and in the right hand side we need to find out the value 1 by x marginal cost minus average cost which is 1 by x. Marginal cost is b plus 2 cx minus average cost that is a by x minus b minus cx. Here we have the derivative of the average cost function as minus a upon x square plus c which can also be written as c minus a upon x square. And here we have 1 upon x 2 cx minus cx will give us cx minus a upon x that is equal to c minus a upon x square. So here we have left hand side equal to right hand side therefore we can write that the derivative of the average cost function is equal to 1 by x the marginal cost function minus the average cost function. So this ends the session hope you understood it well with the help of the cost function first find out the marginal cost function and the average cost function then substitute its value in the left hand side as well as right hand side. You will find out left hand side when equates with right hand side then you will be able to prove the given equation. So this ends the session have a nice day.