find the maximum value of $$xy + yz +zx$$given that $x+2y+z=4$  my attempt : $(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx) $   or $2S=2(xy+zx+zy)=(x+y+z)^2 -x^2-y^2-z^2=(4-y)^2-x^2-y^2-z^2$  $2S=-x^2-z^2-8y+16=-x^2-z^2+4x+4z$  from the the above we can say due to symmetry maximum value occurs at $x=z$  hence $S=-x^2+4x$ whose maximum is 4  is this right or/and is there a better way ??
if $x+2y+z=4$ and $x,y,z$ are real number. then find maximum value of $xy+yz+zx$  putting $x+z=4-2z$ in $y(x+z)+zx = y(4-2z)+zx = 4y-2yz+zx$  i wan,t be able to  go further,could some help me with this
Let $\displaystyle f$ be differentiable, $\displaystyle f(x)=0$ for $|x| \geq 10 $ and $g(x)=\sum_{k \in \mathbb Z}f(x+k).$ How does the clean Installation of Windows 10 after the free upgrade work? I have not found any information from MS - only speculations from people like me, so I hope someone here has had better findings.  When I upgrade a qualified version of Windows to Windows 10 and then want to do a clean install, I've heard that the installer won't ask for a key. But what if my HDD is formatted before I attempt the clean install? How does Windows 10 try to detect the valid "upgraded" status? It seems that it doesn't in such a case, at least as far as I've tested.  So my question is: In that (or a similar) scenario: How do you clean install Windows 10?  In my case it's actually even more complicated. I'm from Germany so I can re-use OEM-licenses due to law. I have two qualified Windows OEM licenses which I both installed in a VM on my Linux installation and upgraded both to Windows 10.  Now I want to install one of them onto my bare physical machine - how would that be possible? Is there an official way of doing this or do I need to wait for those unofficial "backup license"-tools being updated to work with Windows 10? What would you call a word that doesn't exist in or translate well into another language? Turn off close Facebook friend notification No more updates for Google Chrome 32-bit I am using Google Chrome 48.0 32-bit. Today when I launched Google Chrome, I got this message:    This computer will soon stop receiving Google Chrome updates because this Linux system will no longer be supported   I thought Ubuntu 14.04 is supported for five years, what's exactly happening here?  And on 64-bit Ubuntu, I get an error from Update Manager and apt-get update:  Failed to fetch http://dl.google.com/linux/chrome/deb/dists/stable/Release  Unable to find expected entry 'main/binary-i386/Packages' in Release file (Wrong sources.list entry or malformed file)   Some index files failed to download.   They have been ignored, or old ones used instead. How can I know which radio button is selected via jQuery? Adding coordinate system to layout in QGIS composer? What is the remainder when $2^{1990}$ is divided by $1990$? I actually do not have the basic idea on how to approach these type of questions....so please tell me a generalized method about all this too.  It came in RMO, and the question is:  What is the remainder when $2^{1990}$ is divided by $1990$ ? Alphabetically display the items in itemize How to determine which window manager is running Is there any way (such as a command or environmental variable) to determine what window manager is the one actively running in the current session? "He didn't know where New Jersey was" I know the past tense carries the past tense in every dependent clause, but referring specifically to places or to things that are eternal, like the Earth, seems a bit weird and therefore we sometimes (I believe incorrectly) say     He didn't know that New Jersey was actually on the East Coast.   Because it still is. Or     He thought the Earth was round.   So is it square now?  Logically speaking, would you consider the use of past tense here a bit confusing in a day-to-day speech in these examples? Would you instinctively opt for using the present tense? Regular (non-moderator) users should be able to override deletions by Community user There is a question on  that got deleted by the Community user. This question isn't terrible (we have worse one's there), and I'm not complaining about this question in particular.  I wanted to vote to undelete it. However, it says "A moderator has deleted this post and it cannot be undeleted". This "moderator" is Community!  I understand if a "real" live human moderator (no offense to all non-humans out here) deleted this question that we can't override it. But this is a Robot! It sees that it got a downvote, no answer and little activity for a while and just deletes it.  Can we have "Community user" have rights to delete, but have those "deletions" subject to normal-person override? brute force password guessing on SSH server isn't going to work? I'm running an SSH server on my personal computer. The log had many people(or bots?) repeatedly trying to log in to my server (that is before I changed the default port), which made me a little freaked out. I was worried about their dictionary attacks or brute force password guessing.  But then if you type wrong password (or wrong username), the server waits for about 2 seconds before it asks for password again. So an attacker can try one password per 2 seconds. So I concluded that these attacks aren't that great a threat (as long as your password does not consist of dictionary words.) Am I correct? I want a single word embedded in whole logo. How do i do it in adobe illustrator?  I want to design a logo like this. Kindly help me What is this story about 20 y.o. inexperienced "kids" fighting against a stronger foe in space? Why use the 'ref' keyword when passing an object? I already derived a QM expectation value for ordinary momentum which is: $$ \langle p \rangle=  \int\limits_{-\infty}^{\infty} \overline{\Psi}  \left(- i\hbar\frac{d}{dx}\right) \Psi \, d x $$ And I can read clearly that operator for momentum equals $\widehat{p}=- i\hbar\frac{d}{dx}$. Is there an easy way to derive an expectation value for $\langle p^2 \rangle$ and its QM operator $\widehat{p^2}$? How are ETF prices calculated after the market is closed For example:  Because the NASDAQ: DAX is tracking the index INDEXDB: DAX should its trading value be completely flat when the market the markets are open in the US, but closed in Germany? I'm not seeing that artifact in any of the charts I can find. Volume of the solid generated by revolving the region R enclosed by the curve - Disk and Shell method Find the volume of the solid generated by revolving the region $R$ enclosed by the curve $y=x^3$, the $x$-axis and the line $x=1$.  (About the line $y=1$ using the disk and shell method.) Yesterday, one of our computers dropped to grub shell or honestly, I am unsure what shell it was when we turned on the machine.  It showed that it can't mount the root filesystem or something in this sense, because of inconsistencies.  I ran, I believe:  fsck -fy /dev/sda2   Rebooted and the problem was gone.  Here comes the question part:  I already have in her root's crontab:   @reboot /home/ruzena/Development/bash/fs-check.sh    while the script contains:  #!/bin/bash touch /forcefsck   Thinking about it, I don't know, why I created script file for such a short command, but anyways...  Further, in the file:  /etc/default/rcS   I have defined:   FSCKFIX=yes    So I don't get it. How could the situation even arise?    What should I do to force the root filesystem check (and optionally a fix) at boot?  Or are these two things the maximum, that I can do?  OS: Linux Mint 18.x Cinnamon 64-bit.  fstab:  cat /etc/fstab | grep ext4   shows:   UUID=a121371e-eb12-43a0-a5ae-11af58ad09f4    /    ext4    errors=remount-ro    0    1    grub:   fsck.mode=force    was already added to the grub configuration.