if $a,b,c &gt; 0$ then prove that $abc(a+b+c) \le a^3 b + b^3 c + c^3 a$  My attempt The hint given for this question was to use the Cauchy-schwarz inequality. But if you look at the expression given, the L.H.S is symmetric but the R.H.S is not. If I understand the inequality(Cauchy-Schwarz)  correctly, both the sides of the inequality should be symmetric. So can we use the Cauchy-Schwarz to prove this? I also tried applying AM &gt; GM but got stuck. Any help would be appreciated.
Let $x&gt;0$, $y&gt;0$ and $z&gt;0$. Show that   $$x^3y+y^3z+z^3x\ge xyz(x+y+z).$$   I known we can't WLOG: $x\ge y\ge z$, if this, I can use rearrangement inequality, But other I can't it. Thanks?
What nutrients can humans absorb in the mouth? For instance, I realise being able to absorb simple sugars in the mouth is pivotal in the rapid action of oral glucose gel. Thus I was wondering what nutrients in general can be absorbed directly within the mouth, and at what speed? Where has the trailing newline char gone from my command substitution? If $[\cdot]$ denotes greatest integer function, then what is the value of natural number $n$ satisfying the equation $$[\log_21]+[\log_22]+[\log_23]+\dots[\log_2n]=1538 ?$$   My try:  Note that  $$0+1\times2+2\times2^2+3\times2^3+4\times2^4+5\times2^5+6\times2^6+7\times2^7=1538$$  But the answer is $$n=\sum_{i=0}^72^i=255.$$  How is it derived? Android "M" was officially announced on at the Google I/O conference on May 28, 2015 and revealed to be 6.0 Marshmallow on August 17, 2015.  You can review the high-level changes at the  and the announcement on the official .  Of course, not all devices will get this update right away and others won't get it at all. Each manufacturer and carrier often choose to add their custom modifications which tend to delay the official release.  When will each device get Android 6.0, either officially or unofficially, if ever?  See also: Install Microsoft Office Pro 2010 on Ubuntu 13.04 What is the difference between the Bash operators [[ vs [ vs ( vs ((? I am a little bit confused on what do these operators do differently when used in bash (brackets, double brackets, parenthesis and double parenthesis).  [[ , [ , ( , ((   I have seen people use them on if statements like this :  if [[condition]]  if [condition]  if ((condition))  if (condition) Placing items before \documentclass When looking in books or at sample code, I always see \documentclass appearing first. Is that absolutely necessary?  I have created some test code which seems to compile fine:  \def\SetClass{article} \documentclass{\SetClass} \begin{document}     This is some text. \end{document}   Is there any problem with placing other items, such as \input, \usepackage, or such details before the \documentclass? At most finitely many (Hamel) coordinate functionals are continuous - different proof I need to make this equation in LateX but I get some errors when i try to do that.    Can you help me please?? If $f(x)=x+\int_{0}^{1}[xy^2+x^2y]f(y)dy$ where $x$ and$y$ are independent variable.Find $f(x).$ Doubt related to a theorem in congruences of Chapter 5 of Apostol Introduction to analytic number theory I am self studying number theory from Introduction to Analytic number theory by Apostol.   I have a doubt in one argument of proof of Theorem 5.14 and I am posting its image.      My doubt is in 2nd line of 2 nd paragraph how Apostol writes - If y is a solution of (5) then ay $\equiv $ at ( mod m) . How does y being a solution of (5) implies that?       Can someone please help Necessary and sufficient condition for the matrix $A = I - 2 x x^t$ to be orthogonal Let $x$ be a non zeo (column) vector in $\mathbb{R}^n$. What is the necessary and sufficient condition for the matrix $A = I-2xx^t$ to be orthogonal? I encountered the integral $$\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx = -0.9393323982...$$ while researching the evaluation of harmonic sums. Mathematica 11 is not able to evaluate this integral, and is not able to evaluate the underlying indefinite integral.  I considered using the Maclaurin series for the expression $\sin ^{-1}(x)^2$ to evaluate this integral. Using this Maclaurin series, it is easily seen that the problem of evaluating the above integral is equivalent to the problem of evaluating the series $$\sum _{n=1}^{\infty } \frac{2^{2 n-1} H_{n-\frac{1}{2}}}{(1-2 n) n^2 \binom{2 n}{n}}=-0.9393323982...$$ which Mathematica 11 is not able to evaluate directly.  I also considered using the Maclaurin series for the expression $\log \left(1-x^2\right)$ to evaluate the above integral. Using this Macluarin series, it is easily seen that this integral is equal to $$\frac{1}{4} \left(16 \pi  C-\pi ^3-\pi ^2 \log (4)\right)+\sum _{n=1}^{\infty } \frac{\, _3F_2\left(\frac{1}{2},\frac{1}{2},1;\frac{3}{2},n+1;1\right)}{n^2-2 n^3}=-0.939332...,$$ but Mathematica is unable to evaluate the above infinite series.  What is a closed-form evaluation of $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$? What techniques can be applied to evaluate this definite integral? Wrong Sentence:     Never before in the history of the world such a thing has happened, I don't think that will ever happen again.   Right sentence:     Never Before in the history of the world has such a thing happened, I don't think that will ever happen again.   Why “has such a thing happened” is the right form, and why can't I use such a thing has happened? Duplicate typeset index entries How to get distance when acceleration is not constant? I have a background in calculus but don't really know anything about physics.  Forgive me if this is a really basic question.  The equation for distance of an accelerating object with constant acceleration is:  $$d=ut        +\frac{1}{2}at^2$$  which can also be expressed  $$d=\frac{\mathrm{d}x}{\mathrm{d}t}t+\frac{\mathrm{d^2}x}{\mathrm{d}t^2}\frac{t^2}{2}$$  (where x(t) is the position of the object at time t)  That's fine for a canonball or something like that, but what about a car accelerating from 0 to cruising speed?  The acceleration is obviously not constant, but what about the change in acceleration?  Is it constant?  I suspect not.  And then what about the change in the change of acceleration, etc. etc.?  In other words, how does one know how many additional terms to add in the series?  $$d=\frac{\mathrm{d}x}{\mathrm{d}t}t+\frac{\mathrm{d^2}x}{\mathrm{d}t^2}\frac{t^2}{2}+\frac{\mathrm{d^3}x}{\mathrm{d}t^3}\frac{t^3}{3}+\frac{\mathrm{d^4}x}{\mathrm{d}t^4}\frac{t^4}{4}\cdot etc. \cdot ?$$ Can I register a retail boxed copy of a game with Steam? I personally do not like to purchase games digitally.  There's just something magical about going to the store, picking up a boxed game, staring at the amazing box art, bringing the game home and putting the disc in the drive.  That being said, I know that several game distribution companies are leaning more toward digital distribution, with[Steam holding a large chunk of that market.  Is it possible to buy a retail boxed copy of a game, but still get the benefits of Steam (automatic updates, install on multiple computers, etc.) ? Deleting my own question Can't get armature to move mesh! WAMP Server not working Okay so a continuation from , where you experts intro'd me to , which can basically execute PHP within a Windows XP environment.  So now I've got it installed, but the tray icon forever shows YELLOW, and when I visit any PHP page in my browser, it just shows me the PHP source!  Also, when I visit "" in IE7 it gives me a 404 Not Found, FF3 just shows a blank page.    BTW I've tried "Restart All Services" and restarting my machine, but it still won't work.  Any ideas? Any of you had this problem and solved it? Please help me here, I'm desperate to execute PHP client-side and I'm just reverting to testing on-server for now!