Earlier we established that profit maximization, Requires, Setting marginal revenue equal to marginal cost. But when we drew our marginal cost curve and our marginal revenue curve, we noticed something. That there's two of those points because the horizontal line, which, of course, is marginal revenue, will hit the u-shaped marginal cost curve on the portion of the marginal cost curve where it's going down as well as in the portion of marginal cost curve where it's going up. And so we have to think here, ask ourselves the question, what really happened here? And so the problem is that, If you think about maximizing profits, you could take a profit function and take the derivative of it and set it equal to zero. And you'd say, well, Larry, that'll help you find the maximum, but it'll also help you find the minimum. So what I'm going to do is I'm going to introduce something here to you. All right, we'll start with a new page. And if I were to tell you that you had some general function that looks like this and it's y = f(x). Some general function, y = f(x), and if you look to points where the slope of the curve is zero, that is it's flat, what you're identifying is a series of peaks and a series of valleys, okay? And for those of you who like calculus know though what you have to do, those are called local min and local max. And you have to take a second derivative test to find out whether you found a minimum or a maximum, whether you found the peak or a valley. I'm not going to make you worry about second derivatives, but I want to show you the intuition of what we're talking about here. And to do that we're going to go back to recalling what our total revenue and total cost curves look like, so let me draw this axes. And on the vertical axis we'll put dollars and cents, on the horizontal axis we'll put quantity. And we had this thing called a total revenue function. And a total revenue function looks something like that, TR. Remember, it's got a constant slope because we are in a situation where it's, let's start with price. We're in a situation where we have some exogenously given price, P0, times whatever output you might want to produce. And then we had this thing called total cost function, and the total cost function had some component that was fixed cost that you incurred even if you didn't produce anything. And then the cost function went up like this, kind of flattened out. And then went up like this. And we've discovered when we did that that, in fact, if I was to ask you to draw me the profit curve, so profit, oops. Suppose I said, show me profit. You'd say, well, Larry, I know that at this output level, We'll call this output level 1, and at this output level, we'll call this output level 2, at both of those output levels profits are zero because total revenue's exactly equal to total cost. So I can map my profit function by finding to zero points. In between those two output points, that is in this region, you can see profits are positive, revenue is in excess of cost. Revenue in excess of cost is a positive profit function, so it looks kind of like this. And for output points in excess of q1, that is output points out in this region, costs are much greater than revenue. And so this profit function is going to be screaming down here negative. Okay, and for outputs less than q2 you can see that, in fact, you're going to have negative profit. This cost function is greater than the revenue, but there's a peak point here, okay? There's a point about right here where they're most, oops, I don't want that, I want that green pen back. And then it's going to hit right here. This'd be equal to minus F0. See, the profit function, if you produce zero output, you get zero revenue, you have zero variable cost, but you certainly still have to eat your fixed cost. So this amount is going to be losses for you if you produce zero, but the peak lost point is about here. Okay, so this output, essentially, This output point maximizes losses or minimizes profits. This output point, Maximizes profits. So this point, q*, maximizes profits, this point, which we'll call, We'll call that q3, that output point, q3, actually minimizes profits [LAUGH] it maximizes losses. And that's the problem is that when you set marginal revenue equal to marginal cost, in this case, marginal revenue go into slopes equal, you maximize the vertical distance here. Over here you're maximizing the vertical distance on the wrong direction. So what we found with this particular problem is that alpha is the profit minimum and beta is the profit maximum. So what we're going to do is we're just going to not do alpha, we're just going to forget that part. And if we step forward, that means the only thing we are going to care about is when marginal revenue equals marginal cost while marginal cost is upward sloping, okay? That's going to be the key part, thanks.