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Find a) any critical values and b) ay relative extrema Ux)=X + 9x-10 8) Select the correct choice below and if necessary fill in the answer box within your choice 0...

Question

Find a) any critical values and b) ay relative extrema Ux)=X + 9x-10 8) Select the correct choice below and if necessary fill in the answer box within your choice 0A. The critical value(s) of the function islare (Use comma t0 separale answers as needed ) The function has no critical values_ b) Selecl the correct choice below and if necessary; fill in Ihe answer box(es) within your choice 0A The " relative minimum poini(s) islare and the relative maximum point(s) is/are (Simplify your answe

Find a) any critical values and b) ay relative extrema Ux)=X + 9x-10 8) Select the correct choice below and if necessary fill in the answer box within your choice 0A. The critical value(s) of the function islare (Use comma t0 separale answers as needed ) The function has no critical values_ b) Selecl the correct choice below and if necessary; fill in Ihe answer box(es) within your choice 0A The " relative minimum poini(s) islare and the relative maximum point(s) is/are (Simplify your answers Type ordered pairs , using integers or fraclionsUse 0 B. comma separale The relalive maximum answers as needed ) point(s) islare and there are no relative minimum (Simplify your answer Type a ordered pair Using points. integers fractions Use comma separate answers a5 The relative minimum point(s) is/are needed ) (Simplify and there are no relative maximum points your answer Type an ordered pair using inlegers or fractions Use There are no comnma separate answers relalive minimum points Js needed ) and there are no relativo maximum points



Answers

a. Find the critical points of the following functions on the given interval. b. Use a graphing device to determine whether the critical points correspond to local maxima, local minima, or neither. c. Find the absolute maximum and minimum values on the given interval when they exist. $$h(x)=(5-x) /\left(x^{2}+2 x-3\right) ;[-10,10]$$

For this problem we are asked to find the critical points of the function H of X equals five minus x divided by x squared plus two X minus three on the closed interval from negative 10 to 10. So our first step is to take the derivative with respect to X. Which we are going to need to apply the quotient rule. So we'll have a negative negative one times X squared plus two x minus three. Then we would have minus five minus X times the derivative of the denominator which is going to be two X plus two. And then all of that will be divided by X squared plus two x minus three. Now we want to solve for when this equals zero which means that we're actually only concerned with when the numerator will equal zero. So we want x squared plus two x minus three Plus 5 -1 times two X plus two. Two equals zero. Where I've multiplied through the numerator by negative one. So we can then also or we then also want to expand out that set of brackets or a second set of brackets. So we'd have 10 x plus 10 then minus two X squared minus two X equals zero. So we'd have x squared minus two. X is going to give us a negative X squared plus two X minus two. X leaves us with actually plus two X. But then there's also a plus 10 x minus two X. It would be two X. Yeah, so we just end up with minus X squared plus 10 X. And then we would have a minus three plus 10. So that's going to be plus seven. We want to solve for when that equals zero. Or alternatively we can write this as x squared minus 10, x minus seven equals zero. Which we can solve using the quadratic formula. I'm going to pause for one so we are going to get to solutions out of this as we would expect one is X equals about negative 1.164. Simple. Check what the boundaries were negative 10 positive 10. So negative 1.164 is our first um route. Then the second route would be At 11.164. So that is outside of our bounds. So If we check we'll see that X equals negative 1.164. It gives us an absolute maximum. Yeah. And then we just need to evaluate at X equals 10. So f of 10 yeah is going to give us our absolute minimum on the interval which is going to come out to let's see here, That comes out to about negative 0.04, one

For this problem in part A We're asked to find the critical points of the function F of theta equals to sine Theta plus coast data on the closed interval from negative to pi to two pi. So to begin, we want to find the derivative of or of F of theta there. So derivative of two. Cynthia is going to go to to coast data and derivative of coast that is going to go to negative sine theta. We want this to equal zero. So in turn that means that we want to our to coast data to equal sign data. Or alternatively, we need to to equal sine Theta over coast data. Which is equivalent to saying that we want to to equal tan theta, which in turn means that data is going to be tan inverse of two, which comes out to be first of all. That is going to be multi valued. So one moment here. So we'll have within our interval There will be 10 inverse of two, 10 in verse of two plus two pi. And then also we'll have 10 inverse of two. 10 Universe of two. Mhm minus pi. Or actually excuse me minus pi. Then Let's see we would also have 10 inverse of two plus pi. So we'll have four different points Where we'll find that at 10 inverse of two. We'll get a maximum and then we'll have We go forward by two pi. That will actually be outside of our bounds, But actually -2 pi is the one that we want there The one with -2 pi then that will give us a minimum. Or actually let me correct myself that's going to be a maximum as well. And then at tan inverse of pie and tan inverse of minus or tan inverse of two minus point, an inverse of two plus pi. We'll get minima where we'll find that F of tan inverse of two as well as at 10 inverse of -2. It will have a actually, I'll be careful here, a 10 inverse of two. It has a value of about 2.236 At 10 in verse of 2 -2 pi Has a value of equal or it has the exact same value 2.236. Then at F of tan inverse two minus pi Has a value of -2.236. And same deal for f of tan inverse of two plus pi. Okay.

Okay, So, looking at our fallen graph, you see that we have an absolute minimum, and this is at this point here instead. That's negative. 0.1 to 5, and then are absolutes. Um, next, that city, that negative four, we get 51 point 23 that's or just fellas, that's at, um X is equal to native for so that's up here somewhere.

So let's start by taking a conservative. So we get to close Sign of data minus side of data, and I'll set this equal to zero to get to Cole Center. Feta. You could see China later. Dividing my coastline gets using detentions. They're so they don't Most people Teoh tangents, inverse of To, which is 1.107148718 Okay, so let's check or critical points. So that's tensions in verse of To as well as our endpoints. Okay, so in this case, we get one for our output. And then what about negative to pine? That's one as well. And then to sign of Tencent, inverse of to plus co sign of tension in verse of to So that gives me 2.24 So I see that this is our absolute max and then we have to absolute men values


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