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either. In this problem, we have actually 2 various plants developing a product and a Q one is the amount created by the first arrangement. You two by the second. So we're going to attempt to figure out the best values if you desire and also que tu to maximize profits. So we need to begin creating a formula for profit. So profit will certainly be this capital P. Here, let's begin by remembering that profit equates to revenue minus prices and additionally more revenue equals This is a little P equates to price, times quantity that you're marketing. So if we simply closely a substitute, we will be okay right here. So we will certainly gain, uh, rather of p best. We have an expression for pee pee is 60 minus 600.4 Uh que Now rather of Q Member que really equates to Q one plus que to miss out on the total quantity, we're told. So that's what I'll carry out below. I'll make this Q one plus cute, as well. Okay, that's that. What we simply did. It's simply pee. So we took treatment of this part Now times Q. And aget. Remember, Key was actually Q one plus que tu such a 2nd check out what taken place here. This initially part is our little P price. Second component is very own Cue the quantity that we're marketing. Okay, Um, that ideal ideal just are our revenue. We market it to subtract off the expense cost attributes. We have actually 2 of them, one for each plant. So what? To subtract off both of them. So they get minus 8.5 minus point. 03 vital one squared, minus 5.2 minus point. 04 Q 2 squared. And so every one of this, it's attracting off price. So that is our prophet function. Have to clean it up a little little bit so that we can execute somepoint through it. And it's a lot of algebra. Um, so, among various other things to be distributing, he desire to ask you to carry out whatever would foil everything out below combined. A couple, choose, terms you through the numbers, so I'm gonna put dot, dot, dot Here. This is the calculus part. That's the tough component Here. I assume that your elder bro's okay, however below you have the right to check your answer versus mind what you have to finish up through, at least perhaps in a different order. But what? You should end up through at the finish of the day is 60 kyu won, plus 60 Que Tu minus 0.7 Key one squared minus 0.8 Q. Two squared on is your appoints Euro Eights Q. One Q 2 minus 13.7 Again. That just comes from algebra arithmetic on this prophet attribute. Now it'll be much easier to take derivatives, though that's the only point. OK, so that's our prophet function. The next point we perform, if you want to find a maximum, is find important points. So let's take partial derivatives. Let's take P sub Q one to the partial derivative of all this through respect to essential one. Okay, this first term will provide us a derivative of 60 following term zero, but there's no essential wanting it the following one. After that, we'll get minus 0.14. Kyu won next term zero cause there's no key one, and also ultimately, the second to last term Que Tu is a constant, so we'll acquire minus 0.8 Q two and also, let's take the partial derivative through respect to cute, too, and also watch what we acquire the initially time that will certainly provide us something is the 2nd 1 right here will provide us 60. Now this fourth term will certainly acquire negative. 0.16. Cute, too. And then lastly, from this term, kyu won this constant. Now some obtaining minus 0.8 Cute, also. That's our cue one. It's essential. It's a cue one. So we have a partial derivatives. Now the whole suggest of doing this. It's a set that equal to zero in and solve. So we set them equal to zero. And what carry out you execute this by hand? Or the graphing calculator? Several means to execute this theater is the system. Their devices of equations. We have actually two equations in two variables. Variables. Air. Just you desire it? You 2 watch the order here. When I was doing this prior to I accidentally switched them. They've acquired the wrong answer the initially time. So make sure you put them in order once you solve ideal. But after you rearrange points, let's see. We must get you all appropriate out this much, so you have to gain somepoint indistinguishable to 0.14 Key one plus allude. Oh, a cute too cool. 60. And over right here, 600.8 Q one plus 0.16 q two likewise amounts to 60. And if we experienced fury multiply via is simply algebra. Um, and the two answers we should gain are Q one equates to 300 and also cute, as well, equals 225. Those are the amounts we need to acquire currently, prior to we complete the problem and also say, We're done, we should check to make sure that we have uncovered a maximum and also not a minimum. All ideal, so these air attend of answers right below, yet all we recognize is that that's a crucial point. We don't understand for sure that it's a max, so inspect appear. We'll do our work examine. We'll require this second partials. So looking at P Q. One appropriate here the derivative of that through respect to Kyu Wan. In various other words, P Q. One Q r. We can see at a glance will get negative. 0.14 Likewise, Peak, You also cute to let's look down below. The extremely bottom, the derivative of peak you to via respect as well cute to is ideal right here. We're gonna youngsters minus 0.16 and also now the blended partial P sub Q one Q to threaten. Look at P Q one best right here. Take its derivative via respect to cute too. And on Lee, the last term will certainly give us something it will certainly provide us minus 0.8 Okay, so we usage all this details for the second derivative test, we calculate our resources D, which we have the right to see the negative 0.14 negative 0.16 minus negative 0.8 squared. Um, okay. And we really don't care around the answer is a lot to be care about its authorize. So if we go ahead and educate this out when you calculate all this out, you acquire a decimal. But even more importantly, your answer will certainly end up being less than zero. He'll gain an adverse answer out of all this, and also that's the important thing. Um, actually here. So if you run the numbers for a second, we gain the right below. And actually D, isn't it? You need to end up with something higher than zero. Again. The number is not as essential as the last scientific research. A D is greater than zero. Let's be clear around that. So Deion's of better than zero. And it's also noticed that peaky one Q one here was much less than zero. And those 2 bits of information, the second derivative test tells us are sufficient to conclude that this instrumental point right here is truly a regional Max.


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So we have actually discovered a maximum key. One must be 300 you 2 should be 225 and