 Hello and welcome to the session. In this session we discussed the following question which says, if the total cost function is given by cx equal to 4x plus 180 and the demand function is given by px is equal to 40 minus x for producing x units of a particular product, find first rate given values, second the values of x that produce a profit, third the values of x that result in a loss. Before we move on to the solution let's discuss some functions related to business and economics that would help us solve this question. First we have the cost function cx which is equal to f plus bx where this f is the fixed cost and bx is the variable cost. Next we have demand function given as x is equal to f of p where this p is the price of the commodity per unit and x is the number of units demanded. We can also denote the demand function as p is equal to f of x. Next we have the revenue function which is rx equal to p into x where again this p is the price per unit and x is the number of units. So we can also say that the revenue function rx is equal to x that is number of units multiplied by p which is the demand function. This is when we consider p equal to fx as the demand function. Next we have the break even point at this point there is no profit, no loss and so in that case the revenue function rx is equal to the cost function cx. This is at the break even point. This is the key idea that we use in this question. Let's now move on to the solution. In the question we are given the cost function and the demand function and it's also given that x units are produced of a particular product. In the first part we need to find the break even values. The cost function cx is given as 4x plus 180 then the demand function is equal to 40 minus x and it's given that x units are produced of a particular product. And so now the revenue function is equal to the number of units produced multiplied by the demand function. So it would be given as x into 40 minus x whole. So we have rx is equal to 40x minus x square. Now first we will find out the break even values. So for the break even values we have the revenue function is equal to the cost function. So this means that 4x plus 180 which is the cost function is equal to the revenue function which is 40x minus x square. And from here we have x square minus 40x plus 4x plus 180 is equal to 0. This gives us x square minus 36x plus 180 is equal to 0. Let us now solve this quadratic equation by splitting the middle term. So we have x square minus 30x minus 6x plus 180 is equal to 0. Further x into x minus 30 the whole minus 6 into x minus 30 the whole is equal to 0. So we have x minus 30 the whole into x minus 60 whole is equal to 0 which gives us x equal to 30 or 6. So these are the break even values that is we can say that for break even the company can produce and sell 30 units or 6 units. In the next part we need to find the values of x that produce a profit. So for profit we have the revenue function rx would be greater than the cost function cx which means 40x minus x square should be greater than the cost function which is 4x plus 180. From here we have 36x minus x square minus 180 is greater than 0. So further we can say x square minus 36x plus 180 is less than 0. Now factorizing this quadratic equation we have x minus 60 whole into x minus 30 the whole this is less than 0. Now we have that x minus 6 the whole multiplied by x minus 30 the whole should be less than 0. And from this diagram we conclude that x lies between 6 and 30. So this means that for profit the number of units produced should be between 6 and 30. We have to find the values of x that result in a loss. Now for loss the cost function cx would be greater than the revenue function rx. This means that 4x plus 180 should be greater than 40x minus x square. And from here we have x square minus 36x plus 180 should be greater than 0. This further gives us x minus 6 the whole multiplied by x minus 30 the whole should be greater than 0. And this inequality would be satisfied only when x should be less than 6 or x should be greater than 30. We conclude this from this diagram. Hence we can say for loss the number of units produced should be less than 6 greater than 30. So this completes the session. Hope you have understood the solution of this question.