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anonymous
 one year ago
NEED HELP! In finding the absolute extrema of the function on the given interval of [pi/4, 7pi/4] when g(x) = x + cot(x/2)
anonymous
 one year ago
NEED HELP! In finding the absolute extrema of the function on the given interval of [pi/4, 7pi/4] when g(x) = x + cot(x/2)

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freckles
 one year ago
Best ResponseYou've already chosen the best response.0So first step is to differentiate to find critical numbers.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0wait, whats the derivative of cot?

freckles
 one year ago
Best ResponseYou've already chosen the best response.0the derivative of cot(x) w.r.t. x is csc^2(x).

freckles
 one year ago
Best ResponseYou've already chosen the best response.0you have to use chain rule there since you have x/2 inside though

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[\frac{d}{dx}\cot(u(x))=u'(x) \cdot [\csc^2(u(x))] \\ \frac{d}{dx}\cot(u(x))=u'(x) \cdot \csc^2(u(x))\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yea I was about to type that... so basically the derivative is g'(x) = 1+ csc^2(x/2) * 1/2

freckles
 one year ago
Best ResponseYou've already chosen the best response.0yeah and we set that =0 to find critical numbers

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[1\csc^2(\frac{x}{2}) \cdot \frac{1}{2} =0 \\ 2 \csc^2(\frac{x}{2})=0 \\ \csc^2(\frac{x}{2})=2 \\ \sin^2(\frac{x}{2})=\frac{1}{2}\] that should make it easier for you to solve that last step was just taking reciprocal of both sides

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0wait...how did you get it to become sin^2 (x/2) = 1/2 ?

freckles
 one year ago
Best ResponseYou've already chosen the best response.0well first step I multiplied 2 on both sides

freckles
 one year ago
Best ResponseYou've already chosen the best response.0second step I added csc^2(x/2) on both sides

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I got everything until the last part

freckles
 one year ago
Best ResponseYou've already chosen the best response.0last step I took the reciprocal of both sides

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so you did: (1/2)*(2) = 1/csc*csc^2(x/2)

freckles
 one year ago
Best ResponseYou've already chosen the best response.0I don't think that is what I did.

freckles
 one year ago
Best ResponseYou've already chosen the best response.0I'm not sure what that is.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0lol could you show me the work for how you got what you got in the last part

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[\text{ if } \frac{1}{a}=\frac{1}{b} \text{ then } a=b\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[\csc^2(\frac{x}{2})=2 \\ \frac{1}{\sin^2(\frac{x}{2})}=2 \\ \frac{1}{\sin^2(\frac{x}{2})}= \frac{1}{\frac{1}{2}} \\ \text{ which means } \sin^2(\frac{x}{2})=\frac{1}{2}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh you just switch out csc to 1/sin

freckles
 one year ago
Best ResponseYou've already chosen the best response.0yes those are reciprocal functions

freckles
 one year ago
Best ResponseYou've already chosen the best response.0csc(u) is the reciprocal sin(u) and vice versa

freckles
 one year ago
Best ResponseYou've already chosen the best response.0which means csc(u)=1/sin(u)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so now we find the critical points.. if its sin(x/2)^2 = 1/2

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0then would it be pi/6?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Oh wait, that doesnt fit the interval

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[\sin^2(\frac{x}{2})=\frac{1}{2} \implies \sin(\frac{x}{2})=\frac{\sqrt{2}}{2} \text{ or } \sin(\frac{x}{2})=\frac{\sqrt{2}}{2}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0how did you get a sqrt of 2 in the second part, wasnt there a 1 there before?

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[\sqrt{\frac{1}{2}}=\frac{\sqrt{1}}{\sqrt{2}}=\frac{1}{\sqrt{2}} =\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2} \\ \]

freckles
 one year ago
Best ResponseYou've already chosen the best response.0i didn't do all those steps I just remember that sqrt(1/2) =1/sqrt(2) or I also know I can write sqrt(2)/2

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yea thats a good thing to keep a note of

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0wait... how do you solve for that x...? I was assuming it could be pi/4, 3pi/4...but it just doesnt look right with how the x is placed in the equation

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[\text{ we have two equations \to solve } \text{ Let } u=\frac{x}{2} \\ \sin(u)=\frac{\sqrt{2}}{2} \text{ or } \sin(u)=\frac{\sqrt{2}}{2} \\ u=\frac{\pi}{4}+2n \pi , \frac{\pi}{4}+ 2n \pi \text{ or } u=\frac{2 \pi}{3}+2 n \pi , \frac{5\pi}{4}+2 \pi n \\ \text{ or condensing this a little } \\ u=\frac{\pi}{4}+\pi n \text{ or } \frac{\pi}{4} + n \pi \\ \text{ now remember we wanted } \frac{\pi}{4} \le x \le \frac{7 \pi}{4} \\ \text{ but we solved for } u \\ \\ \text{ recall } u=\frac{x}{2} \\ \text{ so } x=2u \\ \text{ so we need to find } u \text{ above such that } \\ \frac{\pi}{4} \le 2u \le \frac{ 7\pi}{4} \\ \text{ dividing 2 on all sides gives } \\ \frac{\pi}{8} \le u \le \frac{ 7\pi}{8} \] so before we go further and solve for x can you solve for u on the interval [pi/8,7pi/8]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0wow, didnt think of splitting the equation into 2 separate parts. But yell ill solve for u on the interval

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I dont know how to find where pi/8 and the other value is onthe unit circle

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[\text{ we are solving } \sin(u)=\frac{\sqrt{2}}{2} \text{ or } \sin(u)=\frac{\sqrt{2}}{2} \text{ on } u \in [\frac{\pi}{8},\frac{7\pi}{8}] \\ u=\frac{\pi}{4}, \frac{3\pi}{4}\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.0now since u=x/2 we need to solve: \[\frac{x}{2}=\frac{\pi}{4} \text{ and also solve } \frac{x}{2}=\frac{3\pi}{4}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so x= 2pi/4 and 6pi/4

freckles
 one year ago
Best ResponseYou've already chosen the best response.0reducing x=pi/2 or x=3pi/2

freckles
 one year ago
Best ResponseYou've already chosen the best response.0now you have critical points on the given interval so plug in the critical numbers and the endpoints into f to see which gives you highest output and which gives you lowest output

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0do i plug the x values in the derivative or the original function?

freckles
 one year ago
Best ResponseYou've already chosen the best response.0also for the above equation if it would have made things easy we could have also used the half angle identity for sin \[\sin^2(\frac{x}{2})=\frac{1}{2} \\ \frac{1}{2}(1\cos(x))=\frac{1}{2} \\ \text{ multiply both sides by } 2 \\ 1\cos(x)=1 \\ \text{ subtract 1 on both sides } \cos(x)=0 \\ \cos(x)=0 \\ \text{ this probably would have been easier for you \to solve }\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.0no you plug into the original function the derivative will only tell you if the slope is positive,0, negative ,or nonexisting

freckles
 one year ago
Best ResponseYou've already chosen the best response.0easier in general not just you lol

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[g(x)=x+\cot(\frac{x}{2}) \\ g(\frac{\pi}{4})=\frac{\pi}{4}+\cot(\frac{\pi}{8}) \\ g(\frac{\pi}{2})=\frac{\pi}{2}+\cot(\frac{\pi}{4}) \\ g(\frac{3\pi}{2})=\frac{3\pi}{2}+\cot(\frac{3\pi}{4}) \\ g(\frac{7\pi}{4})=\frac{7\pi}{4}+\cot(\frac{7 \pi}{8})\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.0so you just might want to whip out your calculator and plug those into see which gives the highest and lowest output

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0This problem required a lot of work...I thought it wouldve been easy, I guess I have to review the concept behind this problem

freckles
 one year ago
Best ResponseYou've already chosen the best response.0well I think the only hard part was the way we solved for the critical numbers in the beginning the halfidentity made the equation faster to solve

freckles
 one year ago
Best ResponseYou've already chosen the best response.0half angleidentity *

freckles
 one year ago
Best ResponseYou've already chosen the best response.0but either way will suffice

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Step1 find critical values and replace them in original function Step2 replace endpoints as well Step3 check your highest and lowest values Done :)
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