Find all real number $x$ satisfying $10^x+11^x+12^x = 13^x+14^x$       My Work Dividing by $13^x$ we get  $$\left( \frac{10}{13} \right)^x +  \left( \frac{11}{13} \right)^x +  \left( \frac{12}{13} \right)^x = 1 + \left( \frac{14}{13} \right)^x$$ The LHS is a decreasing function of $x$ and the RHS is an increasing function of $x$. So there is only one intersection in their graph. I am looking for a formal way to find the root. I know that $x=2$ works. But how to formally find this root?
Find all real numbers $x$ for which $$10^x+11^x+12^x = 13^x+14^x.$$   I thought about taking the logarithm of both sides, but didn't see how that would help. How can we make the equation simpler?
Blog showing up as mobile, won't exit How to automatically resize the vertical bar in a set comprehension? I have a simple command which I use to write set comprehensions so that the braces automatically resize with respect to size of the enclosed arguments, e.g.  \usepackage{amsmath} \newcommand{\set}[1]{\left\lbrace #1\right\rbrace}  \begin{equation*} \set{x \in X \mid x &gt; \frac{1}{2}}   \end{equation*} % Produces something like: { x : X | x &gt; 1/2 }   However, the \mid vertical bar wont resize like the braces { .. } do. How can I get the \mid to resize as well? Homeomorphism between the unit disc and the unit square I know that the function that describes the homeomorphism is: $f(x,y) =       \begin{cases}     \frac{x^{2}+y^{2}}{\max(|x|,|y|)}(x,y) &amp;\quad\text{if } (x,y)\ne  (0,0) \\       \text{} (0,0) &amp;\quad\text{if } (x,y) =  (0,0) \\       \end{cases}$  But it's impossible to me to show that $f(x,y)$ is an homeomorphism. Mostly i can't see the injectivity. indefinite article before proper names Kafka, discovered in the 1920s by the Expressionists, damned by the Nazis in the 1930s and by the Marxists at more or less the same time, was reinvented after World War II by the French Existentialists. They saw in him the father of Angst and the Angst for the father: an Albert Camus before his time.   Can you tell me what the purpose of using the indefinite article before the proper name is in the sentence above. What it be a mistake to write just "Albert Camus" without the article? I am using an APEX batch to process and insert large amounts of data. Essentially I would like it to process every row to catch any errors with the data and then rollback the entire batch if a single error is found. Is there any simple way to do this?  I have tried using a Save Point, but because they are not serialisable, it doesn't work across the entire batch.   Just to clarify, by error I mean an error that I have defined, error with validation, etc. Not an Exception thrown by the system. Is there a rule of integration that corresponds to the quotient rule? Can every game of Klondike-Solitaire be solved? I play a lot of Solitaire games on my Android phone and love to keep an eye out for the statistics.  Given that the Solitaire version lets you restart the game endlessly, I usually play until I solve it. But I never managed to solve more than 80% of the games played (1000+).  So now I wonder, is every Solitaire game solvable? So I have these two wheels parented to the leg of a robot. I wanna make them spin a few times but they're spinning around the wrong axis. With other objects I could just press R, Y, Y to make them spin right when they're already at an angle, but for some reason it doesn't work with these. Would be amazing if someone helped me fixing this! How do I disable DVD auto eject in Windows 7? I just discovered an incredibly annoying "feature" of Windows 7 tonight.  If you click on your optical drive in Explorer, and there is no disc in the drive, Windows 7 is kind enough to open the tray for you.  I'm finding it hard to contain my rage over this idiotic design decision.  WTF?  According to , this behavior was apparently introduced in Vista.  The author posted a workaround, but I don't want a workaround.  I want it to stop doing this.  Like he, I have a case with a door that covers the drive bays.  If it's closed, and I accidentally click on that optical drive, it knocks on that door 10 or 12 times before giving up.  Can this behavior be turned off in Windows 7? Word for person always after what he does not have and not cherishing what he has Possible Duplicate:          Is there a word for a person who is always after what he does not have and does not cherish what he has? If not, any idiom or phrase? Ignore folders/files when Directory.GetFiles() is denied access I am trying to render this scene above [Snap-shot 1]. But when i am trying to render the scene, its showing me the result [Snap-shot 2]. Whats wrong with my blender [2.72] Proof of Wolstenholme's theorem I know someone or the other has asked about disabling ⌘Q, but I like the shortcut, and I use it often. However, I use an email client called Airmail that unfortunately does not keep a service (for notifications) running in the background when it is quitted. However, if I press the close button a service runs in the background, allowing me to receive notifications.  But since I have a habit of pressing ⌘Q all the time to close apps, I was looking into a way I could disable it for only Airmail.  Anyone know of a what to do this? How do I get attention for old, unfixed bug reports and feature requests without official responses here on Meta? I posted a question here on Meta Stack Exchange, or on a per-site meta, regarding a bug I found in the system. I got no comments, or a few comments stating that this is a legitimate bug (and not by design). However, it's been quite a long time, and no official response was ever posted there.  Or, I had an idea for a new feature, but realized it's already requested, either by searching or posting a request myself only to have it closed as a duplicate of the existing one. It's been a while since the original request was filed, but no official response has been made.   I'm aware that staff members are , but either my post isn't about one of the indicated topic areas for which they're accepting escalation requests for that month, or I already flagged to request escalation but it was declined, so the instructions there don't apply for my post.  What can I do to draw attention to meta questions for the SE team, so they can give an official response, and possibly implement the bug fix or feature?  Note: This is different from other general questions which ask for how to get attention for unanswered questions, since the procedures on meta sites are different, and getting an official response isn't necessarily the same as getting a good answer. What free Remote Desktop (server) solutions are there? I know Ubuntu comes with a "Remote Desktop" option that appears to be a straightforward VNC server, and I'm trying to understand the alternatives. Here are the possibilities I've heard about so far:   VNC  VNC + SSH Tunnelling NX Server, free edition FreeNX NeatX X2Go X11 Forwarding over SSH xrdp    I'm coming at this from a Windows user's perspective: To the best of my experience, RDP (aka Terminal Services) is a reasonably secure (barring mitm/server spoofing), efficient desktop sharing protocol with well-supported clients, that can be exposed to the internet when necessary without major fears of intrusion. To the best of my knowledge straight VNC is none of those things, which is where I get confused - why wouldn't a better desktop sharing technology be developed or used in the open-source world? I know VNC can be wrapped with SSH, but that seems beyond the reach of a casual user. X11 forwarding over SSH may be more or less efficient, I have no idea, but is definitely even more complicated, and doesn't (as far as I know) give you access to already-running stuff (no desktop sharing as such, just remote application running).  So, I'd like any feedback/preferences amongst these or any other "Free" desktop sharing options, using these criteria and/or any others:   Security (esp. for access across internet) Efficiency (bandwidth usage, responsiveness, etc) Free-ness, as in Speech (not sure where RDP or FreeNX lie for this) Free-ness, as in Beer (are there any commercial solutions with usable dependable free offerings?) Ease of use (server and client side) Cross-OS Client availability Cross-OS Server availability Support for independent sessions and shared (and/or "Console") sessions Ongoing support/maintenance/development   Thanks! I know base 2 and base 10 , MB and MiB, but i'm not understanding the byte counts shown for a directory, by du and by windows explorer The du command shows 43G in use. I'm not sure whether that's base 2 in the sense of 2^30 or if it's base 10, in the sense of 10^9.  But anyhow -   Running this du command from cygwin, to see the size of the windows directory (done from an administrative command prompt - since it seemed to have some issues reading some directories otherwise)  user@comp ~ $ du -sh /cygdrive/c/windows 43G     /cygdrive/c/windows   Windows Explorer  which I understand shows it in 'base 2', (a larger unit for KB,MB,GB than base 10 KiB, MiB e.t.c.), and hence a lower quantity of that larger unit is required to give the same result.    Windows explorer shows      46.6GB  , over 50 billion bytes.    (according to windows explorer)  or  46.8GB  , over 50 billion bytes. (according to windows explorer)  So, du aside.  Even if I multiply the larger figure, 46.8GB  by 1.048576 then I get 49.0733568 that's not over 50 billion. So that's part A of the question,  but then also, part B of the question, why is the du figure so low. How can I copy from Tmux terminal and paste it other application in my OS? I am using tmux on ubuntu. I read this  for copy and paste inside tmux. But I would like to know how can I copy from TMUX terminal and paste it to other application e.g browser in my OS? Ever since upgrading to Windows 10 1903 the other day, my CPUs have all been pegged at 100%.  "Antimalware Service Executable" is using 60%, and "Microsoft OneDrive (32 bit)" is using 35% (with the other 5% being spread amongst everything else running).  After running like this for a day or so my computer case (Surface Tablet) is smoking hot.  How can I find out what's causing this problem?  Is there logging I can review for those services?  Known issues with 1903 that I just can't find on Google? Why didn't the Emperor realize that torturing Luke to death in front of Vader might be a bad idea?