Find the maximum of this expression $\frac{3}{a^3+b^3 + 3} + \frac{3}{b^3+c^3 + 3} + \frac{3}{c^3+a^3 + 3}$ , where $a,b,c $ are real numbers such that $abc = 3$. What I have done is to use Cauchy-Schwarz Inequality to obtain: $\left(\frac{1}{a^3+b^3 + 3} + \frac{1}{b^3+c^3 + 3} + \frac{1}{c^3+a^3 + 3}\right) \left((a^3+b^3+3) + (b^3+c^3+3)+ (c^3+a^3 + 3)\right) \geq (1+1+1)^2 \geq 9$ Then we would have: $\left(\frac{1}{a^3+b^3 + 3} + \frac{1}{b^3+c^3 + 3} + \frac{1}{c^3+a^3 + 3}\right)\left(2(a^3 + b^3 + c^3) + 9\right) \geq 9$ $\left(\frac{1}{a^3+b^3 + 3} + \frac{1}{b^3+c^3 + 3} + \frac{1}{c^3+a^3 + 3}\right) \geq \frac{9}{2(a^3 + b^3 + c^3) + 9}$ By AM-GM inequality we have: $ a^3 + b^3 + c^3 \geq 3\sqrt[3]{a^3b^3c^3} $ $ a^3 + b^3 + c^3 \geq 3(abc) $ $ a^3 + b^3 + c^3 \geq 9 $ . . . I have gotten $\left(\frac{1}{a^3+b^3 + 3} + \frac{1}{b^3+c^3 + 3} + \frac{1}{c^3+a^3 + 3}\right)\geq \frac{9}{2(a^3 + b^3 + c^3) + 9} \leq \frac{1}{3}$ $\left(\frac{3}{a^3+b^3 + 3} + \frac{3}{b^3+c^3 + 3} + \frac{3}{c^3+a^3 + 3}\right)\geq \frac{27}{2(a^3 + b^3 + c^3) + 9} \leq 1$ After this, I am unable to break flip the inequality sign and obtain $ \leq $ sign instead in order to obtain the maximum. Much help appreciated
if $a,b,c$ are non zero positive reals prove $$\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$$ I have used A.M  G.M  inequality as follows:  $$a^3+b^3+c^3 \ge 3abc$$ adding $abc$ both sides we get  $$a^3+b^3+abc \ge 4abc-c^3=c(4ab-c^2)$$ so  $$\frac{1}{a^3+b^3+abc} \le \frac{1}{c(4ab-c^2)}$$ But since $a,b,c$ are positive reals  $$4abc-c^3 &lt; 4abc$$ so  $$\frac{1}{4abc-c^3} \gt \frac{1}{4abc}$$  but i am unable to proceed here
"Its" as a Possessive Pronoun Since its can be both determiner possessive pronoun and nominal possessive pronoun, an example of its as determiner possessive pronoun would be:     We saved this question for last because of its complexity.   Because complexity is a noun, so its must be a determiner possessive pronoun in this sentence. I'm looking for an example where its is used as nominal possessive pronoun in a sentence as above. Has the speed of light changed over time? Could someone judge my (stoner) hypothesis that the speed of light has changed over time -- i.e. as the universe has expanded in volume light has slowed down, perhaps going so far as back to the big bang when it was infinitely fast and there was no time because everything happened at once etc. Thinking that the speed at which information can propagate through the universe is linked to the size of it seems intuitive to me. My question -- is there an easy disproof of this? Would Einstein have to be wrong? Does it violate anything supposedly more fundamental such as quantum or string theories? Do any current experiments invalidate it? If not can you show me in any case why you think its unlikely.   Edit 8/24  I'm accepting Mark M.'s answer but will post this here because there is a character limit on comments  @Mark M thanks, good answer but as someone who has only read some popular physics, and should leave this to the experts, I'm still muddled in my personal theory. I don't see why you should need two units to measure the speed of light. The thing I have a hard time wrapping my head around is the relation of time and distance. They seem like they could fundamentally be the same thing. If you say time is measured fundamentally by the vibration of so and so quantum object in space... why can't we just measure that vibration distance as the constant... I'll repeat myself to try to be clear... there is a certain minimum distance that particles have to go to interact with each other... if it wasn't vibrating there wouldn't be time,its what creates the illusion of time...so instead of talking about speed or c as distance/time.... can't we simply talk about that distance a quantum object vibrates....I'll lead up to my point.....perhaps there can need be only one constant here, and that is the physical size of the universe. a tiny metal tuning fork doesn't appear to be vibrating at all but if you blew it up to the size of the empire state building the metal rods would move from window to window. perhaps as our universe expanded in size the length of that minimum vibration (perhaps infinite at point zero) would have expanded, and therefore created the illusion of time and the speed of light, which, as the universe expands, will continue slowing down. perhaps we are like a big balloon and we have been blown up and all the fields/particles-without-size are vibrating more and more in that space.. am i missing something obvious here? Why do some devices have 200V on the case? I recently got a desktop LED lamp, and I feel a "strange" slight vibration when touching the metal case. I found that other devices also have high voltage on their case.  What is the reason/trade-off to design the PSU that way? Are "real" transformers inefficient? Even if it's harmless, why is the high voltage connected to the metal case, and not just through insulated wires?    When I turn the lamp on, the setup in the picture measures the power line (220V) 214Vpp (59Vrms). When I also touch the metal case, the voltage drops down to around 30V.  AC adaptor: How do I revert a Git repository to a previous commit? Database general error: 2006 MySQL server has gone away Area 51 Site is quite different from other SE sites and needs an SE footer Honestly, navigation through Area 51 is a bit confusing and seems to be much different from the other SE sites.    The feature that is very lacking is the footer with all the sites you are subscribed to so that a user can easily return to a different SE site. This one really bugs me.  Can this be easily fixed?  Also, unlike all the other sites there is no "meta" site to add comments specifically about that site.    Again Area 51 seems out of sync with the other SE sites. How can I test for a player wearing specific armor? How can I test for a player that is wearing an iron helmet named "test1" and an iron chestplate named "test2" in Minecraft Version 1.12? Equivalent conditions for a preabelian category to be abelian Let's fix some terminology first. A category $\mathcal{C}$ is preabelian if:  1) $Hom_{\mathcal{C}}(A,B)$ is an abelian group for every $A,B$ such that composition is biadditive,  2) $\mathcal{C}$ has a zero object,  3) $\mathcal{C}$ has binary products,  4) $\mathcal{C}$ has kernels and cokernels.  A category $\mathcal{C}$ is abelian if it is preabelian and satisfies:  5) every monomorphism is a kernel and every epimorphism is a cokernel.  Define the coimage of a map to be the cokernel of its kernel, and the image to be the kernel of its cokernel. We have the following commutative diagram:    where $\overline{f}$ is the only existing map (because of universality of kernel and cokernel).  I'm having trouble proving the following:     A preabelian category $\mathcal{C}$ is abelian iff $\overline{f}$ is an isomorphism.   The converse is easily shown, I'm having trouble proving $\Rightarrow$... Distribution of the maximum of two correlated normal variables Edits on closed posts Should questions or answers on questions that are closed be edited, even if the question is closed and the edit does not change anything about the reason it's closed? Example:  was just edited, which I saw in the review queue. It certainly improves the answer, but since the answer doesn't really have to exist on SE at all (because the question is off-topic), should this edit be made? In my understanding, the only reason that the edit option for a closed question is not greyed out is so that it can be edited to make it comply with the SE rules. How do World of Warcraft referrals and the free month work? I sent the referral link to my boss and he bought the game and applied his CD key.  He now has a paid account linked to mine:    Yet when I view my World of Warcraft account, it seems they didn't give me the free month. It appears I only have the free month that came with my Battlechest purchase.  I just created and bought my account this morning. Would this affect the free referral month in any way?    We haven't seen each other or partied online in the game yet, do we need to do that first in order for me to receive the month? Is using passive voice "bad form"? removing event listener I have a CSV file which has postcode data in a field, but it is non-spatial e.g. no Easting and Northing.   I have a shapefile with postcode in a field and some additional data.  I want to join the CSV with the shapefile based on attributes (postcode fields). I know I can do this in ArcMap, but how can I achieve the same thing in QGIS? ASP.NET MVC Custom Error Handling Application_Error Global.asax? Area enclosed between the curves $y=x^2$ and $y=60-7x$ Find the area enclosed between the curves $y=x^2$ and $y=60-7x$.  I am completely new at this but I have tried and I believe that it should be between the numbers $0$ and $60$ is this right? The Monty Hall problem Intuitive explanation of entropy I have bumped many times into entropy, but it has never been clear for me why we use this formula:  If $X$ is random variable then its entropy is:  $$H(X) = -\displaystyle\sum_{x} p(x)\log p(x).$$  Why are we using this formula? Where did this formula come from? I'm looking for the intuition. Is it because this function just happens to have some good analytical and practical properties? Is it just because it works? Where did Shannon get this from? Did he sit under a tree and entropy fell to his head like the apple did for Newton? How do you interpret this quantity in the real physical world? I know we can use both Present Simple and Present Continuous to describe future actions. But I don't know when to use the correct form.   For example:  We fly to Paris next week.  We are flying to Paris next week.   These sentences both seem correct to me.  Can anyone explain me how to correctly use these tenses? Finding critical points (extrema) directly without using eigenvalues If we have a function $f: \mathbb R^3\rightarrow \mathbb R$ of the form:  $$f(x)=(a_1x_1^2+a_2x_2^2+a_3x_3^2)e^{-x_1^2-x_2^2-x_3^2}$$  where $a_1&gt;a_2&gt;a_3&gt;0$.   We can find the critical points by:  $$\frac{\partial f}{\partial x_1} =2x_1e^{-x_1^2-x_2^2-x_3^2}(a_1-a_1x_1^2-a_2x_2^2-a_3x_3^2)=0$$  And solving symmetrically for $x_2$ and $x_3$. We get critical points: $(0,0,0) , (\pm 1,0,0) , (0,\pm 1, 0) , (0,0,\pm 1)$   We could determine the nature of them by looking at the eigenvalues of the Hessian, but is there a way to just analyse it directly? Obviously $(0,0,0)$ is a strict global minimum, but I can't quite see that the other points are. Any pointers/hints would be appreciated.