For this problem. We're considering a TV manufacturer. We're gonna look at a few functions that describe its procedures. Each week is demand function or price function, revenue function and profit function. But let's start just by what we know about this manufacturer. They sell TVs, So let's let X equal the number of TVs that they sell in a given week. So P of X is going to be our price function or our demand function. That is the price per TV. Now we know something about how the price and the number correlate to each other. We know that if X is 1000 when they sell 1000 TVs, they do that when the price point is $450. We also know that if we get a $10 rebate, so if the price goes down, the number of TVs we sell goes up, go decreasing. The price by 10 means an increase of 100. So assuming that this correlation is linear, we confined the slope of this line. We take the difference of our exes. Are differences sorry difference of our wise. So as before 40 minus 4. 50 over the difference of our X is 1100 minus 1000. That gives us negative 10 over 100 which matches what we're told in the problem. When the price goes down by 10 the number goes up by 100 is an inverse relationship. So it is a negative slope, and I can say that that's equivalent to negative 1/10. Okay, so I know the slope. I know points that I know two points so we can use the point slope form of a line which says why minus y one equals m times X minus X one. And we can also do a rewrite here. I can move that. Why one over to the right hand side and say that Why are output equals M times X minus X one plus why one? So let's do that. I'm going to take the slope, which we have right here. Negative 1/10 and I'm going to take a point. It doesn't matter which one. I'm gonna take the first one just because it's first. You'll get the same answer no matter which point you use. And my equation is why my output, which in this case I can call p of X. That's my price function or my demand function equals m my slope times x minus X one. In this case, that's 1000. Pull us Why one which in this case is 4 50. And if I want to get rid of those parentheses, that's negative. 1/10 x negative, 1 10 times negative. 1000 is plus 100. Add that to the 4 50 I get 5 50. So that is my demand function or my price function. Now. My next question is, I want to maximize revenue. That's a good thing most companies dio. So if I want to maximize my revenue, what's my price point? How much of a rebate should I offer? Well, if you recall, revenue can be found by taking the price per unit times the number of units sold. So if you have ah 100 units sold and they sell for $100 each 100 times 110,000, that's your revenue. So we're going to multiply these Well, we just found P of X. That equals negative 1/10 X plus 5 50 times X. And if I want to get rid of those parentheses. That's negative. 1/10 X squared plus 550 x. There's my revenue if I want to maximize it. Remember, optimization problems we have are function. We're going to take the derivative. We'll set that equal to zero that will give us any optimal points that we might have. So if I take the derivative, that's gonna be negative. 2/10 or negative 1/5 X plus 550. If I set that equal to zero. What that gives me is 550 equals 1/5 X or X equals Oh, I just wrote that down X equals 2750. So that's how many TVs we need to sell in order to maximize my revenue. What's my price for those? I'm gonna come up here at the top. I've got a little more room, my price, or to my demand function. When X is 2750 I'm gonna plug in 2750 here, and that's going to give me a price of $275. So what rebate gives me that price? Well, if I'm starting at $450 if I want to subtract $275. That means I need to have a rebate of $175 in order to maximize the revenue I'm bringing in. Okay, Last question we have We've only talked about money coming in. This is how much each one sells for its how money we sell. This is our revenue. Incoming money. What I truly want to maximize is my profit money in minus the money out because it costs something to make these TVs. So I'm gonna be maximizing not my revenue, but my profit. And since I already have a p function, I'm gonna let f of x b my profit. This will be my profit function. Profit function is going to be revenue minus cost revenue. I found from right here we already found our revenue function. I want to subtract my cost function, and we're told that our cost function it's 68,000 plus 1 50 x. Now I'm going to just combine some things here. I'm gonna combine my like terms. This is negative. 1/10 x squared. I have plus 5 50 x minus 1 50 x. That's gonna be plus 400 x minus 68,000. So there's my profit function toe. Maximize my profit function just like with the revenue. I'm going to take the derivative and we'll set it equal to zero. So this gives me negative 2/10 or negative 1/5 X plus 400 setting that equal to zero. I get 400 equals 1/5 x by multiplied by five. I get X equals 2000. I'm gonna go down just a little bit of kind of run out of room here. Where's my Christian? Okay, so come down just a little bit. So if Access 2000, what I want to know is, what's my rebate I have to offer? So what's my price? Well, my price function when X is 2000, that's going to be negative. 1/10 times X plus 5000 or 550. Okay, that gives me a price of $350. That's my price. What rebate do I have to offer? Well, if I start with 4. 50 and I want to have 3. 50 is my final price, it means I have to offer a rebate off $100 to maximize my profit