Okay, So we'll be looking at some problems that combine both linear and quadratic functions, particularly with respect to manufacturing. And the 1st, 1st example we're gonna look at has to do with uh, wristwatches. So a company is selling making and selling wristwatches and there's what's called a revenue function. That just means the amount of money that they make on the rift, not profit, but the amount of money that they bring in the income relative to the risk, the wrist watches. And that revenue function is 75 x minus 2/10 X squared. And then we also have a cost function that is the cost to make the wrist watches. And so when we look at those two graphs together, then the the if we're looking first of all, let's just look just at the revenue function. So if we want to find out how many wristwatches we need to sell in order to maximize revenue without any regard to cost, then we're gonna maximize revenue right here At 187.5. So obviously we can't sell half a wristwatch. So our revenue would be maximized at either 187 wristwatches or 188 wristwatches. Both of those would give us maximum revenue. So it's gonna max out at 187 or 188 revenue maxes out At 187 Or 1 88. And that maximum revenue would be whatever value we get when we substitute 187 or 188 into The function. So it would be, we're gonna take 75 multiply that by 187 -2/10 Times 187. And that gives us a maximum revenue of $13,987 and 60 cents. Oh no, that's not right, I didn't square that. It's 75 times 187 minus 2/10 Times 187 Squared. Here we go. That's better. Um and that's a maximum revenue of $7,031.20. I knew that it wasn't right because it didn't agree with. There wasn't close to that Vertex maximum there. So when we sell 187 or 188 Wrist watches, we have an income, just an income, not a profit of an income of $7.31, 20 cents. Now, the profit function, if you think about this realistically a profit function is just gonna be the cost that it takes to the revenue, the total that you bring in for the sale of whatever product or selling minus the cost to make that product and that's going to be called the revenue. The profit function. So the profit function is going to be simply the revenue, the amount that you bring in minus whatever it costs you to make that product. So when we substitute These in, we get a profit function of 75 x minus two tents, x squared- your cost function of 32 x plus 1750. And then when you distribute that negative sign, you get a profit function, you distribute the negative sign and you re arrange your variables to make to write your expression in standard form, you get negative 2/10 X squared plus 43 X -1 750. So that's going to be your profit function. And then we'll look at our profit function in order to maximize profit. So then when we graph the profit function, it's gonna look like this here, this purple graph And clearly we're gonna have a maximized profit here at 107 five wristwatches. So profit is maximized on either side of that profit is maxed At 107 or 108 watches. And that maximum profit Is going to be. So we're just gonna take p of 107 and that gives us negative point to times 17 squared Plus 43 times 1 7 -1750. And that gives us a profit of $561.20. Now notice it's not exactly 561 25 because that's if we were just at 107 and a half watches, which we clearly can't do. Um so your maximum profit is also going to be the same at 108 because remember that axis of symmetry of your parabola goes straight down the middle. So these two points on either side of the vertex are going to be um reflections of each other across that That axis of symmetry. So your profit is maximized at 107 or 108 watches for a maximum profit of $560.20. And then if you think about it, um why is the profit max? Why are these two answers different? So why is the revenue max? And the prophet max is different? Well, clearly it's because the revenue max doesn't take into account the cost to produce the material or to produce the product. In this case the wristwatch.