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click here for the question

I tried to prove it using the convexity

MOUNO ANIMATIONS
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    See https://brilliant.org/wiki/jensens-inequality/ – TheBestMagician Jan 04 '22 at 19:17
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    Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Shaun Jan 04 '22 at 19:44

1 Answers1

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Let $$x=a+b+c$$ and consider $x$ as fixed, then we want to find the largest value of $$\sin a+\sin b+\sin c$$. This value occurs when $a=b=c$. That follows from the inequality

$$\sin a+\sin b=2\sin \frac{a+b}{2}\cos \frac{a-b}{2}\leq 2\sin\frac{a+b}{2}$$ which means that if two are unequal we can increase the value by replacing them with their means and keeping the sum constant. From this the inequality follows.

Rene Schipperus
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    This is correct and elegant. I wish you had waited to answer until the OP edited the question to show some work of their own. – Ethan Bolker Jan 04 '22 at 19:59
  • @EthanBolker I think I would have had to wait a long time :-) – Rene Schipperus Jan 04 '22 at 20:05
  • True. So the OP wouldn't get your help. But I think the OP hasn't earned that help until they try first. – Ethan Bolker Jan 04 '22 at 20:14
  • @EthanBolker Maybe next time OP will post a better question. – Rene Schipperus Jan 04 '22 at 20:21
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    I agree with Ethan Bolker: we must do the maximum to avoid helping people who haven't made a single effort. – Jean Marie Jan 04 '22 at 20:26
  • @JeanMarie The milk of human kindness flows freely on this site. – Rene Schipperus Jan 04 '22 at 20:27
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    This is not a problem of kindness: there are a lot of cheaters, unfortunately, who ask questions on this site. Some of them even in ongoing tests... – Jean Marie Jan 04 '22 at 20:31
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    @JeanMarie Those people should take care to state in the question when the exam period will end so we can be sure to submit our answers on time. :-) – Rene Schipperus Jan 04 '22 at 20:36
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    I'm not sure it's kind to answer so quickly, even if it's not an exam or homework question. If you wait the OP might learn more than they would simply reading your answer. Particularly if it's as nice as this one. – Ethan Bolker Jan 04 '22 at 20:41
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    @EthanBolker Yeah, but I am trying to get my answer in before someone else answers it, so this waiting scheme doesnt make sense to me. – Rene Schipperus Jan 04 '22 at 20:43
  • So you have a different goal here than helping people learn mathematics ... – Ethan Bolker Jan 04 '22 at 20:49
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    @EthanBolker Maybe I want to be the FIRST person to help them, like an early responder. Anyway, what is your goal in being here ? – Rene Schipperus Jan 04 '22 at 20:56
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    My goals are to help people with mathematics, have fun and keep myself mathematically engaged in my retirement. I admit to some reputation vanity, but don't pursue it. – Ethan Bolker Jan 04 '22 at 21:00
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    @EthanBolker If that is so, why do you care what other people do ? It does not affect you. – Rene Schipperus Jan 04 '22 at 21:06
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    Last comment. It affects my goal, indirectly, by short circuiting learning for some OPs. Finally, I've looked at your profile. You do write really nice answers. – Ethan Bolker Jan 04 '22 at 21:12
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    ok last comment, how do you KNOW what other people need ? How do you know what is best for THEM ? Dont I get the right to answer the questions I thinks are interesting and to share the insights I have ? Maybe it helps OP, maybe not, but I cannot really know either way. And maybe someone else sees it and they like it. Besides cant I be a reputation hound also ? :-) Good night. – Rene Schipperus Jan 04 '22 at 21:22