Gaokao Math-22, Jiangxi Province 2008:  Define a function $$f(x)=\frac{1}{\sqrt{1+x}}+\frac{1}{\sqrt{1+a}}+\sqrt{\frac{ax}{ax+8}}$$ Prove that, for all positive real number $a$, $1&lt;f(x)&lt;2$.  I tried differentiating but the derivatives got too messy. Tried applying AM GM inequality too but in vain. I believe there lies a more elegant solution to the problem. I' m out of ideas. Please also mention how you came up with the solution.
Let $a&gt;0$, show that for $x&gt;0$, $1&lt;f(x)&lt;2$, where $$f(x)=\frac{1}{\sqrt{1+x}}+\frac{1}{\sqrt{1+a}}+\sqrt{\frac{ax}{ax+8}}$$  I could take the derivative, find the maximum of the function and conclude. But that involves heady algebra and you have to solve a high degree polynomial. Is there an easier way simply using inequalities? Thanks, much appreciated
"I had my house [be] burned down" I have found out that using the verb be in passive constructions such as:      I had my house be burned down    is incorrect, therefore it should be      I had my house burned down.    But is it possible instead of be to use get? E.g.      The wire is passed through the pliers in a specific way to avoid having it get bent.   The rope had its surface wear.   The rope had its surface get worn.   The rope had its surface worn.   The first two sound very odd to me, but just for clarification could you please point out the errors and which ones are correct? Or is using get, be it in -ing or the to infinitive, in any situation along with have incorrect? What's wrong with this .desktop file? Create a workflow rule when Lead status is changed from X to Y I want to create a workflow rule which should be triggered when I change the Lead Status value from 'Open: Not Contacted' to 'Open: Working'  Any help with this please? What is duplicate content and how can I avoid being penalized for it on my site? This is a general, community wiki question regarding duplicate content.  If your question was closed as a duplicate of this question and you feel that the information provided here does not provide a sufficient answer, please open a discussion on .     What does Google consider to be duplicate content? Will the way I am presenting my content result in a duplicate content penalty? How can I avoid having my site's content treated as duplicate content? join comma delimited data column Editing default custom object home screen I would like to add a custom button (URL to Flow) to my default home page for my custom object. Currently, I have a default "New" button to create a new record (see attached screen shot). I'm wanting to add my Flow custom button to this page. Is there a way to do this?  Cheers, Sean. What is the structure in "as best you can"? I instinctively translate it "as best as you can", however this makes no sense.  What is the real structure behind this phrase?  I'll include an  to illustrate the historical presence of this phrase:    Whatever the structure is, I'm guessing that it has something to do with 19th century grammar. Proving the sum of two independent Cauchy Random Variables is Cauchy How can I use "." as the delimiter with String.split() in java When should \cdot be used to indicate multiplication? Is this purely personal style, or are the generally accepted guidelines? Clearly, when dealing with numbers either $2 \cdot 2$ or $2 \times 2$ is needed, but within formulae where an absence of an operator implies multiplication, when should a \cdot be used? I tend to use it only sparingly if it aids in grouping. (Of course, barring the cases where it's required, e.g. a dot product.) How to disable autorefresh in snap Is it possible to disable autorefresh in snap? This  says that Snaps are updated automatically in the background to the latest version, every day. Some snaps (e.g. vlc) are built daily, and it makes no sense to install them every day. Guarding against cryptanalytic breakthroughs: combining multiple hash functions Show that, in any finite graph with at least two vertices, there must be two vertices with the same degree.  HINTS ONLY! When I entered the -al attribute with the  command, I got the following:  anupam@JAZZ:~/blog$ ls -al total 76 drwxrwxr-x 13 anupam anupam 4096 Aug 28 21:45 . drwxr-xr-x 34 anupam anupam 4096 Aug 28 20:52 .. drwxrwxr-x  8 anupam anupam 4096 Aug 28 12:19 app drwxr-xr-x  2 anupam anupam 4096 Aug 28 12:19 bin drwxrwxr-x  5 anupam anupam 4096 Aug 28 12:19 config -rw-rw-r--  1 anupam anupam  154 Aug 28 12:19 config.ru drwxrwxr-x  3 anupam anupam 4096 Aug 28 12:37 db -rw-rw-r--  1 anupam anupam 1339 Aug 28 12:19 Gemfile -rw-rw-r--  1 anupam anupam 2871 Aug 28 12:19 Gemfile.lock drwxrwxr-x  7 anupam anupam 4096 Aug 28 21:45 .git -rw-rw-r--  1 anupam anupam  466 Aug 28 12:19 .gitignore drwxrwxr-x  4 anupam anupam 4096 Aug 28 12:19 lib drwxrwxr-x  2 anupam anupam 4096 Aug 28 12:29 log drwxrwxr-x  2 anupam anupam 4096 Aug 28 12:19 public -rw-rw-r--  1 anupam anupam  249 Aug 28 12:19 Rakefile -rw-rw-r--  1 anupam anupam  478 Aug 28 12:19 README.rdoc drwxrwxr-x  8 anupam anupam 4096 Aug 28 12:19 test drwxrwxr-x  6 anupam anupam 4096 Aug 28 12:33 tmp drwxrwxr-x  3 anupam anupam 4096 Aug 28 12:19 vendor   What does each column mean, and what does the total 76 signify here? How does the loopback interface work I know from reading about the gravitational waves detected by Ligo, that when an object has angular acceleration, it produces gravitational waves.  I'm wondering if an object creates gravitational waves when only accelerating in one direction, however?   I'm also curious as to how the nature of the waves would differ in this case. Is there a way to set my current location manually on Windows 8? Is there a way to set my current location manually on Windows 8? I don't have a GPS module, but the device is always located at the same place. The automatic detection via network sets a place 200 kilometers from here.  I searched for a setting that allows me to set the location by hand, but I didn't find one. Subdomain versus subdirectory I recently moved from having a main site like www.example.com with a subdomain associated site yyy.example.com, to moving everything off the subdomain to a subdirectory like www.example.com/yyy/, with the same hierarchy underneath the subdirectory, and now yyy.example.com is just a redirect page to the subdirectory, so yyy.example.com/abc/page-x is redirected to www.example.com/yyy/abc/page-x and so on.  The effect has been that the traffic increased by several times (the total page views are more than ten times at the present rate) what it was to the yyy subdomain. Previously almost all of the traffic to the yyy subdomain was from the www site. This is from monitoring the old yyy site for a year or more.  I'm curious to know if there's any research or results which would indicate whether this is universal (subdirectories beat subdomains) or "just me". Limit of $\lim_{(x,y)\rightarrow(0,0)}\frac{x^2y^2}{x^3+y^3}$ I want to know whether $\lim_{(x,y)\to (0,0)}\dfrac{x^2y^2}{x^3+y^3}$ exists or not. I tried to approximate to (0,0) from different "paths" and the result was always 0. For example,  $f(x,mx^2) = \dfrac{m^2x^3}{1+m^3x^3}$  But that doesn't show that the limit is 0. I'm using terminal-ide  which is basically a terminal for android with all sorts of utilities. One is able to create bash scripts for example. Is there a way to create an icon on the home screen that will launch one of these scripts?