 Recall that profit is the difference between revenue and cost and actually represents the amount of money earned by production and sales, right? So suppose we have a cost function of the following function, following commodity we're producing, right? So our cost is 2,100 plus 0.25X. So how we wanna interpret this as the following, production of this item that we're gonna sell has a fixed cost of $2,100. That might include like the money we pay for our employees on a given day. That might be the cost of running machines, electricity, land usage, there's whatever, right? There's some cost that's gonna be that on a given day to run our factory, it's gonna cost $2,100 just to run it. But then each item we produce will be a quarter, 25 cents. So we get this nice linear cost function. There's a fixed rate plus the fixed cost plus the rate of actual producing. And our factory can produce at most 30,000 items in a single day. We can go anywhere from zero to 30,000. That's the cost associated to it. Suppose that the revenue comes from the following here, 2X minus one over 25,000X squared. Revenue is commonly gonna be a quadratic function because revenue in general is gonna be a product of X times P where X is the number of things you produce to P is the price you sold it for. Well, if the price function is itself a linear function, which is very common, if you have linear demand, your price function, which is its inverse, will be a linear function. And therefore X times a linear function will give you a quadratic revenue. And so then profit, the profit function we can compute by taking the difference of these things. So profit P of X in general is the revenue minus the cost, which if we take the revenue function, 2X minus one over 25,000X squared and subtract from that 2,100 plus a quarter X, this would give us our profit function. Combine like terms, we would end up with our profit being negative 2,100 plus 1.75X minus one over 25,000X squared. And so this is our profit function. So if we wanna figure out what is the maximum profit that we can make by selling this building, producing and selling this thing, we wanna find the maximum, right? So the maximum is typically gonna be at the vertex. The vertex in this situation, because if you look at your parabolas graph, you're gonna see something like the following. So the points of most interest would be like the X intercepts. Now for profit, this is gonna be pretty lame because that would be your profit is zero. You made no money. This one over here would be like you produce nothing. So you make no money. This profit over here would be that when your cost and revenue cut even, so you actually make nothing. But ideally there's this point at the vertex right here where cost is not too high and we make money from sales that we can actually make a profit. The difference between the two things is positive. So using our formula for the vertex, negative B over two A, we see that the optimal production level is going to be negative 1.75 divided by two times negative one over 25,000. Which that's the same thing as 1.75 times 25,000 divided by two. Anyway, you don't feel free to use a calculator to help you out here. You end up with 21,875. This is how many products, how many objects you should make, how many widgets you should be making in the factory, which is a whole lot smaller. I mean, not a whole lot smaller, but it is within the range of the 30,000 we saw earlier. This will be the best number to produce because this is where cost and demand kind of meet each other at this nice optimal location. Now that tells us how many we should produce in our factory but what's actually the maximum profit? The maximum profit P, we have to plug in 21,875 into our function and then compute it negative 2100 plus 1.75 times 21,875. Try not to round until the very end, minus 21,875 squared over the 25,000. Again, use a calculator to help you out here. There's all this arithmetic that is not super necessary for us here. We have a lot of analysis that we really are trying to understand here. And then when you crunch that through your profit machine, we would have a profit of $17,040 and 63. So this would be our estimated profit from producing about 22,000 little widgets in our factory on a given day. And so this gives you just a taste of some of the applications we can do with quadratic functions. When it comes to quadratic functions, we're often interested in optimizing things. What's the best, right? What's the best price? What's the best production level? What's the biggest garden? Just to name some of the examples we've done in this lecture. The best, the optimal, the maximum, the minimum usually are corresponding to the vertex of the parabola which we might want the X coordinate or the Y coordinate context will tell you which one it is. Now there are some situations where the X intercepts are considered the best but oftentimes when we're looking at a quadratic we care about the vertex and we care about the X intercepts. Maybe the Y intercept as well. And those are the values we care about the most when working with quadratic functions.