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mathmath333
 one year ago
Find the minimum and maximum value of
mathmath333
 one year ago
Find the minimum and maximum value of

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mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0\(\large \color{black}{\begin{align} \dfrac{x^2x+1}{x^2+x+1},\ x\in \mathbb{R}\hspace{.33em}\\~\\ \end{align}}\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0It's hard to explain...but I will see if I can show the work.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0are you talking local or absolute maxes and mins?

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0i would also prefer a way without calculus ( idk calculus)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Do you need to show your work for this problem?

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0not necessary i should understand the idea

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Based on what I know algebraically , I got a minimum of (1, 1/3) and a maximum of (1,3).

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0u calculated the coordinates where as it should be the numerical value

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ok. Let me try a different way.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ok. I got the minimum is 1/3, and the maximum is 3.

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0how did u calculate

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\frac{x^2  x + 1 }{ x^2 + x + 1 } = 3, when (x) = 1\] I got this after simplifying to figure out what x = for the numerator

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Same for the denominator, except it is when x=1

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0It's not the way I was taught, but my aunt is a math teacher, so she taught me her way of solving these pesky sorts

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3eh i like your questions lol

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3make me struggle :)

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0i have tons of question but no time

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3i cheated haha min=1/3<y<3 let's how we get to this

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0one way seems to put random \(x=\pm 0, \pm 1,\pm 2, \pm 3\) but doesnt work always http://www.wolframalpha.com/input/?i=table%5Bf%28x%29%3D%5Cdfrac%7Bx%5E2x%2B1%7D%7Bx%5E2%2Bx%2B1%7D%2C%7Bx%2C10%2C10%7D%5D

Empty
 one year ago
Best ResponseYou've already chosen the best response.1Honestly just graphing this is the best way to do this lol.

Empty
 one year ago
Best ResponseYou've already chosen the best response.1That's why Descartes invented graphing in the first place, to make stuff like this easier. And then we got computers, lol.

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3i suggest that we study \[y=1+\frac{2x}{x^2+x+1} ~~x\in \mathbb{R}\]

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3minus not plus mistake haha

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3\[y=1\frac{2x}{x^2+x+1}\]

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3hmm need to find upper and lower bound m<y<M some how i think that finding the range of x for 2x/x^2+x+1 might be our way!!!

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3I'm just throwing stuff as i go lol

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3x^2+x+1=0 has no real solution yes? so this parabola is >0

Empty
 one year ago
Best ResponseYou've already chosen the best response.1We can factor it and for just simplicity of writing I'll make this: \[\omega = e^{i 2 \pi / 3}\] \[\frac{x^2x+1}{x^2+x+1}=\frac{(x+\omega)(x+\omega^2)}{(x\omega)(x\omega^2)}\]

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3y1=2x/x^2+x+1 can we find the range for this going by what i just did?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0let \(x=\tan z\) with \(z \in (\frac{\pi}{2}, \frac{\pi}{2})\) you will get\[f(\tan z)=\frac{\tan^2 z \tan z+1}{\tan^2 z +\tan z+1}=\frac{4}{2+\sin (2z)}1\]

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3hmm interesting @mukushla

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3i feel that my way can be a lot easier haha though don't you think

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3just need to find the range for 2x/(x^2+x+1)?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah, making a perfect square in denom, that's a lot better than mine

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3need pen and paper!

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0i like that tan substitution looks something creative

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0let me continue with yours\[f(x)=1\frac{2x}{x^2+x+1}=1\frac{2}{x+\frac{1}{x}+1}\]actually there is no need to make perfect square

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0we just need to note that\[\leftx+\frac{1}{x} \right \ge 2 \]

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3hmm seems you a different approach

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3i was trying to use parabola and eyeball the range

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3how is x+1/x>=2

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3first time i see that :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0AMGM inequality\[x+\frac{1}{x} \ge 2\sqrt{x.\frac{1}{x}}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0if \(x<0\) it will be\[x+\frac{1}{x} \le 2\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0equality occurs when x=1 and x=1 for two cases

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3so for x=1 we have x+1/x<2==> x+1/x+1<1 then y<3

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3for x=1 by the same reasoning 1/3<y

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3excellent @mukushla

Empty
 one year ago
Best ResponseYou've already chosen the best response.1That's a nice trick with that AMGM inequality.

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3beautiful indeed :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@mathmath333 ty, actually it is a nice trick to use trig sub for finding range, you may try this one\[f(x)=\sqrt{x2}+\sqrt{4x}\]

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3hmm if one realize they can use trig sub the problem becomes really easy since sin and cos are easily bounded

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0the range of x here is \(2<x<4\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0when you wanna do a trig sub, find the domain, it will give you the clue for the form of sub

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3well domain above is R how can that help

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3but tan has pi/2 singularity?

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3does not matter ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0no we will treat bounds in limit, so it doesn't matter

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0now, what about the second function?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0note that\[2 \le x \le4\]is a restricted domain

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3yeah just about to say that math did it already

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3so what sub is that telling us to use?

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3sin or cos ?

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3hold on don't tell me yet

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3hmm no really because they are both have same bounds 1 and 1

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yes and because\[ 1\le x3 \le 1\]

Empty
 one year ago
Best ResponseYou've already chosen the best response.1They're both the same function just shifted argument by 90 degrees. \(\cos x + 3\) will work just as well as \(\sin x + 3\)

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3so in that we put x3=sinz yes?

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3damn i feel weak in trig lol :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh yeah, \(3+\cos z\) will be better maybe

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3hmm you said they all work fine No? okay lets go with x=3+cosz

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0well derive with cos will be easier

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3so \[y=\sqrt{\cos z+1}+\sqrt{1\cos z}\]

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3so \[0\leq y \leq 2\sqrt{2}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh no you can't just plug in there values, your trig function is not alone, you'll need more simplification

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3if we simplify we just get radical 2?

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3if i didn't make an error that is

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[f=\sqrt{2} \left( \left\sin \frac{z}{2} \right+\left\cos \frac{z}{2} \right \right)\]

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3cosz=2cos^2x/21 oh i see i threw the square when i did my simplification haha

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3i didn't i mean*

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok, good discussion, tnx guys

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3well isn't that the same result above? 0<y<2root2

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3thanks to you learned bunch of stuff here :)

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.3this one i never remember \[\frac{a+b}{2}\leq \sqrt{ab}\]
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