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Find Fhe extremizer 0 f EUu,UJ= SS Ilux#Uy_)dlxdy Sbied +o SS_udxdy 0 iS Gdomus in 1he YY_plum...

Question

Find Fhe extremizer 0 f EUu,UJ= SS Ilux#Uy_)dlxdy Sbied +o SS_udxdy 0 iS Gdomus in 1he YY_plum

Find Fhe extremizer 0 f EUu,UJ= SS Ilux#Uy_)dlxdy Sbied +o SS_udxdy 0 iS Gdomus in 1he YY_plum



Answers

Find $f^{\prime}(0)$ if $$ f(x)=\left\{\begin{array}{ll} g(x) \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0 \end{array}\right.$$ and $$g(0)=g^{\prime}(0)=0$$

Here we have capital f of X. It's a composite function. It has three layers. We've got the F on the outside, We have the three F in the middle and then we have the four F on the very inside and our goal is to find capital F prime of zero. So we have to start by finding capital f prime of X. So using the chain rule, we start with the derivative of the outermost function. So we have f prime of its inside three f of four f of X. Now we multiply by the derivative of the middle function, the three F function. So the derivative of three F would be three times f prime. So we have three times f prime of its inside for f of X. Now we need to multiply by the derivative of the inside function and the derivative of four f of X would be four f prime. Okay, Now, we're not gonna worry about simplifying that right now because our goal is to find the value at zero. So let's go ahead and substitute a zero in there at this point. Sweet F prime of three f of four f of zero times three f prime of four F of zero times four of prime a zero. Now we can take the values that were given. F of zero is zero weaken substitute that in in two places and f prime of zero is too. And we can substitute that in one place. And now we have f prime of three F of four times zero times three f Prime of four times zero times four times two. Okay, let's clean that up a little bit. It's we have f prime of three F of zero times three of prime of zero times eight. Now we can go ahead and find f of zero again and substitute that End that zero and f prime of zero. Substitute that, and that's going to be too. So we have f prime of three times zero times three f Prime of zeer Looks that when we know that one was to prime of zero was too. Times eight. Okay, so we have f prime of zero times three times two times eight and that would be times 48. Now, one more time. F prime of zero is too. So with two times 48 so we get 96

We're given the F double part of excess equals vaccinated to power after one is a Syrian after which is a good sir. So start off with lost find a problem x probably X is equal to And so we have X to the negative one. So we haven't see negative one over X plus c and then every find off our backs, you get negative natural log of X who is the sea ex? Post e. So now we can plug into different pieces of information that we know. We know that when effort want half of one is you go to South Africa to is equal to zero. So what? We can try plugging out in to see what you get. And we also know So because we have half of what is to go to zero and after eight years ago, they're now that half of one is able to achieve. So it's for started out. So we have zero is equal to negative natural log. Absolutely. I leave one plus C terms one post e. You can't actually simplify that down to zero. No, it's just right is equal to so we don't the natural all of one is just Sarah. So we have C plus D is equal to zero. Next we'll smell. Got zero sequel to natural log after about Chu plus see terms to Post E which will get us zero is equal to natural walk of shoe Plus to see plus de so for you, Lina Moxley. A zero is equal to see Post E No, he's so trucked. You look it. Zero Is he cool too Negative. Natural log of Chu course See, eso we get See is people to the natural log of chip. And so now we just need to figure out d Yes, so we know that. See Post e you go to zero. If we have see being the natural go to Post E is equal to zero for a D plus equal to the natural log of the negative Wanted to look up to south see any back into our original question which is up here We'll get f of X equal to the negative natural log of absolute value of X applause The natural walk 02 x minus The natural log

Okay, The first thing we know we can do is we can take the first integral, which means our new exponents becomes to the negative one. We're dividing by the new exponents negative one, which gives us one over negative x pussy. So now we know that we can plug end to get our constant C and deep. Remember, the negative natural law of one is just zero because the natural of one is just zero, therefore got cancels. Therefore, we end up with d equals negative c. This is important because now we're doing half of twos. Negative, naturally of two plus two C plus d, which gives zero is negative. Natural log of two pussy which gives us the natural log of two equals c right, because we're transferring this over to the left hand side. Therefore, changing a symbol which gives our final equation as affects is negative not for a lot of acts, plus Ellen of two x minus natural order, too

Segment is if and then with US Ex egos. X Q minus. Q. Rex Bless one on if off. No, they're Castillo equals one. If off d'oh equals Gino. So this is the given statement. Therefore, we'll find if that's X, which is equal to extra power for a pawn four minus two into extra Squire upon to plus X plus. She won aunt because we know that it does. And you know he waas geo class Seaman one equals C one. Therefore, we have agency one as one. No if X immigrants, when upon for into X to the power of fire upon fire minus x Q upon three plus experts quiet upon to glass See evil ex glass See you. And if off geo equals C two therefore Jew equals C two No, the update and Stephen as one and see you as to the four f off does X because it was to the power for my food. My nose. It is quite less thanks plus well, uh, off x the guns Extent About five bomb 20 minus themself cube on three plus extra Squire upon Do plus thanks plus C two, which is Joe. Therefore, the what answer is this one


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