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$$\begin{array}{l}{\text { (a) Find the intervals on which } \mathrm{f} \text { is increasing or decreasing. }} \\ {\text { (b) Find the local maximum and minimum values of } \mathrm{f} \text { . }} \\ {\text { (c) Find the intervals of concavity and the inflection points. }}\end{array}$$
$$f(x)=e^{2 x}+e^{-x}$$

Can I point to problem number two That C equals F X y equals 12 X square minus 16 x y plus nine y square five following using the former definition of the partial derivatives. Okay, so that's because the former definition of the partial derivatives we have to do, uh, limit as close to zero. And if is f sub x x y you have f x plus age. Why minus f x y over age. So that's just, uh, first right. Is this formula Hair f sub x x y is Yeah, he was limited. H goes to zero of x plus h y minus of x y our h That was our function. Our function F x y is 12 x square minus 16 x y plus nine y square. Now we just need to plug in those values So we've got limited age goes to zero now x f x plus age. We just substitute Exhale by and plus H square minus 16 X plus age. Why plus nine five square minus F X y is just this part 12 X square. Now we've got a plus 16 x y minus nine by square over age Okay, Um, let's see, this part is canceled. Now is this part is 12 x square plus 12 H square for loss two X H minus 12 24 X h Um, subtract this part is 16 y 16 x y and subtract 16 h y mm. So we can cancel this 12 X square and 16 x Y 16 x y minus. So in the new manager, we just have 12 a square plus 24 x h, yeah, minus 16 h y over h equals 12 age plus 24 x minus 16 y sorry limit. And she goes to zero this part and since age because zero this part goes to zero. So this name and is just 24 x subtract 16 y That's for problem A. We want dizzy over DX. The answer is 24 at minus 16. Y you can check it by just, uh, differentiate dysfunction. Just with respect to X, that's the definition. Um, you take y as a constant. So this part goes to zero. This part of the 24 x you use power rule, this part is just 16 y minus. So yeah, the same answer. But you have to do it by the former definition here. Okay, so that's problem. A problem The we want to know Dizzy over dy at the same time, we have to use the formal definition. So formal definition here is similar to X. But now this one is with respect to why? So we have, um, limit edge goes to terror of X. Now here we have y plus age minus f X Y. Thank you. All right, are function f x y. That's what I did. Here is 12 x square minus 16 x y plus nine y square. So this part, we just plug in, we got limit edge goes to zero. Top part is going to be 12 x square x Stay the same but we got to change y two y plus h plug in this part plus nine y plus h square minus ethics wise is part 12 X squared minus minus plus 16 x y minus line y square over age. So, um, let's see what we got. Hey, she goes to zero 12 x squared minus 16 x y minus 16 X h plus lie y square plus nine h square plus 18 by h to this part. Careful about here. Subtract 12 and square plus 16 x y minus Y y square shoot, shoot cancer. A lot of stuff minus 12 x square by the 16 x y plus 16 x y y y square subtract nine wide square. So our top only going to three terms. So if you divide mhm to eliminate it by age. Here we have negative 16 X here we have by H here we have 18 y. Now it goes to zero. So this particles to zero. So your final answer is gonna be a negative 16 X plus 18 y so you can check it by differentiate this park with respect to y you take X as a constant. So this part goes Azera This party got lack of 16 X right? This party have 18 y So exactly the same answer here. Okay, so that's a problem. The now part C yeah. We want to know d f over d x 51 So this part is just, um f sub x x y says the F O R D X is the same as f sub x x y, and we just plug in X equals five y equals one. Mm hmm. So what do we have here? What's our f sub? X x x y we got it in the part a 24 X minus 16 Y 24 x minus 16 y. Now we're plugging. Actually, it was 5. 24. Multiply five, subtract 16. Multiply y equals one we've got. Okay, this is 1 20 minus 16. So 104 That's for part C. The party is similar. Uh, we want to low f sub y one negative four. So this is just if we got the function f sub y x y we're plugging X equals one white was selected for. So in part B, we got f supply f sub y x y is like to 16 x plus 81 9 to 16 x plus 18 y and we plug in, um, X equals one. Why equals negative four. So we got, like to 16, subtract 72. The answer is going to be elective 88 because that's a twofold problem, too.

We're given the phone information integral from 0 to 9, and I perfects the excess equal to 37. But integral off from zero tonight of G f x d of X is equal to 16 and were asked the soft integral from 0 to 9 of the two types of of X plus three times g of x p x. So start off with we can rewrite into girls. We can rewrite this integral appoint some role as integral nights. Sara of two f of x plus the integral from nights Sara of three dfx. We used to know that we can take out any constant outside of the integral, so we can have to grow nine of their off of our backs cost three times the girl from 09 g of X. And now we actually recognize this because these are the two into girls and we were given at the very beginning so we can just plug it in. So we have two times 37 plus three times 16 which will get us a total of 122 as are integral from there's nine of two effort backs post three g of x dx

Properly. 46. This problem seems complicated, but we became also used. The changes first thing we want to tag, it's it's the point. Here are the same point. So what are is three states minus, while is a point we want we can calculate that axis just a three plus one squirt. So actually, it's just too. And why is minus ni right? Oh, here to minus night. It comes out and we can calculate that Sees a function off. Stop you w is a function off accent. Why an accent? Why are both functions off R s? So we want to use the changes for two times. Right? So let's start. Oh, here. Since G too minus NYSE equals two minus two. And we have ah here crime minus two. So we can calculate that also the same point. So partial the partial Are this close to a partial? See, I should stop you. I shall adopt your time. Particle are right. So the first thing you need to know is are shows the part of you. Oh, since is, uh, he has only one arrival. That is stop you so we can just write easy dw and that is just equal toe crime of you, buddy. Bye. After you. Okay. And what about this? Is that a partial? Are? That's just partial D partial acts. Relax, Michael. R us are so deep partial y. But why partial are I? So the first thing is just to be guided by minus two. Oh, sorry. Law and minus two. Oh, uh oh. Half minus two is five here. Five. Right. The at minus two ways to fight a crime. W is two times a whole thing here. Ah, part of departure lacks is minus one. At this point, times for that special are, uh, no expert to our wares. Very special. Special are all we need to capture it by ourselves. So times r minus impartial are so that's a tool. And there's this and thus parts of G pickle. Why, as a three right and science partner, why partial are wife are stars to r s who are us. And when our is three I say is minus one can catch right this tour five hangs. That's ah, minus one or four. Thus, it's a minus two, three times, two times three. All right, so it's my 18 Whoa. Ah, that's a 72 73. So it's minus one for sakes. Derided by 20 right, But so it's a minus 73 divided by 10. Oh, here's the first officer on the need to keep it a second wife. They're the same partials. The partial acts this equals two. Partial. The part of W and time is part of your party dress, but with the partial ask is part of t our sole x Relax, Socialize thus are so g partial. Why? Why apart less so the first thing in the same for scenes tour fire. I just wore it. Tour I Dying's part. Pretty partial Axe minus one. Push, relax. Socialize home. So it's minus five. Two times are minus as and plus Deepak allies three, right? And our square right when our is three and ask this minus one. You have this to be when there were four. Thus 27 So it's, uh, over 10 plus. Hey, hotel here. The guy won 108 Right. So we have one known I over 10 So that's a second answer. Hope I'm correct.

Well defined problem. We want to consider the intervals on which F is increasing or decreasing. And then we want to find the local maximum minimum values. So we have F of X Because each of the two x plus either the negative X. So with this in mind we want to consider f prime of X. That shows us that the function is decreasing Um from negative infinity to negative .231 and increasing from that point to infinity. Then we see that this is also going to be a local minimum. As a result. Looking at the second derivative, we can see that there are going to be no inflection points and the graph is always concave up. So that's your final answer. And we can compare this to the original graph and see sure enough, concave up, there's going to be one local minimum right here at- .231. And that's it.