If $f(x)$ is a real valued function defined on $[0,\infty)$ such that $f(0)=0$ and $f''(x)&gt;0$ for all $x$.  Consider a new function $g(x)={f(x)\over x}$. I want to comment on the monotonicity of function $g$. I mean I want to find whether $g$ is increasing or decreasing and where.  Here are the things I observed. Don't know if it can help or not.  We can find that $g(0)=f'(0)$  Since second derivative of $f$ is positive, it means $f'$ is an increasing function.   $g'(x)=\dfrac{xf'(x)-f(x)}{x^2}$  But I am not able to proceed.  Thanks for hint.
Problem: Suppose $f$ is continuous for $x\ge 0$, differentiable for $x&gt;0$, $f(0)=0$, and $f'$ is monotonically increasing.  Define $g(x)=\frac{f(x)}{x}$ for $x&gt;0$. Prove that $g$ is monotonically increasing.  Source: W. Rudin, Principles of Mathematical Analysis, Chapter 5, exercise 6.
Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive 0s or two consecutive 1s. Note: Problem from "Kenneth Rosen's DM and it's applications" and solution from "Students solution guide for use with ... applications"  Let P(n) be the number of strings not containg two containing two consecutive zeros or two consecutive ones.  I got the answer  P(n)=P(n-1)+2P(n-2)+2P(n-3)+...+2P(0) +2  but then the book says the sequence also satisfies the recurrence relation  P(n)=2P(n-1)+P(n-2). I am not able to understand this part. I have noticed that the smackpacks say on the package that the package is meant for a 5 gal. batch and that you should pour the pack directly into the wort.   On the otherhand - nearly everyone tells me to always make a starter.   Is this just because making a starter never hurts or does the package contain wrong instructions?   I would have guessed that the yeast producers would tell you on the package if you should need some extra steps to get a good product. recovering deleted file held open by apache? Suppose an apache log file gets deleted but it's held open by apache; then this is what I am doing:  pid=$(lsof | grep text.txt | awk '/deleted/ {print $2}') fd=$(lsof | grep text.txt | awk '/deleted/ {print $4}' | grep -oE "[[:digit:]]{1,}")  cp /proc/$pid/fd/$fd directorytobecopied/testfile.txt   This is what I am doing to recover the file and put it back where it was. Is there any simpler way of doing this because above code doesn't look good. Furthermore how do I know from where the file was deleted(directorytobecopied) so that I don't have to manually ask somebody where the file was located originialy and put it back there. How to set the default value of an exposed filter rather than -Any-? I have a view with exposed filters in Drupal 7 and I need to set the default value of a filter to current year. I also need to unset/remove the - Any - option from the select list. A basic grammar rule is to use an instead of a before a vowel sound. Given that historic is not pronounced with a silent h, I use “a historic”. Is this correct? What about heroic? Should be “It was a heroic act” or “It was an heroic act”?  I remember reading somewhere that the h is sometimes silent, in which case it’s an, and when the h is pronounced, it’s a. But then I also remember reading that it depends on which syllable is stressed. And I also think I read somewhere that it might differ between British and American English.  Personally, I pronounce the h, and believe that a is correct. I find that it sounds incorrect to use an and pronounce heroic without the h.  So how do I know when to use a and when to use an with a word beginning with the letter h? Are both acceptable or is there one that is correct? How to use "grep" command to find text including subdirectories What is the difference between "du -sh *" and "du -sh ./*"? What's the difference between du -sh * and du -sh ./* ?  Note: What interests me is the * and ./* parts. Track next Socratic badge progress Just got the  on StackOverflow and, since it can be awarded multiple times, wanted to start tracking progress to the next one, but, the badge is not shown in Available badges to track anymore:    Is it possible to track already awarded badges that can be awarded multiple times? Why isn't Captain America worthy of Mjolnir? I haven't see the movie yet, but in the , we can see Cap trying to lift Mjolnir in the humorous scene where every Avenger gives it a try.  Mjolnir even moves just a tiny bit and at this very moment we see Thor's smile disappearing. This, of course, is intended as a humorous and lighthearted scene.  But it got me thinking: why isn't Captain America worthy of Mjolnir? He seems to be the most "perfect" human being anyone can be: brave, not selfish, legitimate leader, prompt to sacrifice (as seen in the first movie, when he jumps on a fake grenade).  I ask this question in light of this very specific scene, but I'm interested in the Marvel Universe (and Marvel Cinematic Universe) as a whole. Maybe Captain America did wield Mjolnir once? Maybe he has some individual flaw I am not aware of that makes him unworthy? When should I use "shall" versus "will"? Which is the correct use of these two words, and in which context should one be used rather than the other? Suppose I have a directory structure as follows  test test/a test/b   Now I want to execute a command, such that in the . folder I can execute a command on the basename of the files a and b.  So basically, I want something like this, which I naively tried  find test -type f -exec touch `basename {}` \;   Only, this does not result in empty files a and b in the parent directory. Suppose that my touch command can only take a single argument.  This will result in a directory structure like this  a b test test/a test/b   I know how to do this in a bash script, but I am interested in single command solution. I want to get that fancy LaTeX symbol to put in one of my documents but I cannot find it anywhere.  All searching for things like latex symbol have not resulted in the symbol but other symbols. How many of Trelawney's prophecies have come true? If an edit was suggested to an answer to my question and I go to this edit, I see the checkmark This is a little bit off topic as it is not exactly related to TeX.  I saw recently that  was updated to 1.020 which comes with math table support.  Does any one had a chance to play with this font and use it with unicode-math, and in particular with the LuaTeX engine?  I am very interested in purchasing a complete set of this nice fonts family. However I am concerned about its usability with the unicode-math package. Unlike Cambria Math or XITS Math, Minion Math does not come with a single font, instead it has 20 (4 weights 5 optical sizes) fonts. So it will be better to know how it works before spending 700 euro.  In particular, say the following minimal document, will it use correct optical size of the fonts for each of the three formulas? I had trouble with using correct optical size of Minion Pro fonts with fontspec. I worked it out with the SizeFeatures and OpticalSize. However, I am not so sure if the same strategy will work for Minion Math with unicode-math. In addition, in Cambria Math, etc., the bold symbols and alphabets are chosen directly from the unicode math enabled fonts. But in Minion Math, from the glyph tables on their websites, there seems no bold alphabets in each font. So will the mathbf in the document below work out of box?  \documentclass{article} \usepackage{fontspec} \setmainfont{Minion Pro} \usepackage{unicode-math} \setmathfont{Minion Math} % or what ever fonts name actually is  \begin{document} \fontsize{5}{5}\selectfont \[ \mathbf{E} = mc^2 \] \fontsize{10}{10}\selectfont \[ \mathbf{E} = mc^2 \] \fontsize{36}{36}\selectfont \[ \mathbf{E} = mc^2 \] \end{document} OpenLayers 3: Cross-Origin Request Blocked: The Same Origin Policy disallows I have a series of ranges with start dates and end dates. I want to check to see if a date is within that range.  Date.before() and Date.after() seem to be a little awkward to use. What I really need is something like this pseudocode:  boolean isWithinRange(Date testDate) {     return testDate &gt;= startDate &amp;&amp; testDate &lt;= endDate; }   Not sure if it's relevant, but the dates I'm pulling from the database have timestamps. Encouraging people to explain downvotes Site specific magic comment links I would like site specific magic links. Currently you can write [so] in a comment, and it links like this: . I would like to be able to have site specific ones, so for example: English Language &amp; Usage  [dictionary] would go to http://dictionary.com and shows as   Ubuntu  [launchpad] would go to https://launchpad.net/ubuntu and shows as   [launchpad#bugnum] would go to https://bugs.launchpad.net/bugs/bugnum and shows as   [forums] would go to http://ubuntuforums.org/ and sows as   [discourse] would go to http://discourse.ubuntu.com and shows as   [paste] or [pastebinit] would go to http://paste.ubuntu.com and shows as   [install-d] would go to http://ubuntu.com/download/desktop/install-ubuntu-desktop and shows as   [install-s] would go to http://ubuntu.com/download/server/install-ubuntu-server and shows as   ppa:peterlevi/ppa would go to https://launchpad.net/~peterlevi and shows as   There is    Ask Different  [applesup] would go to http://www.apple.com/support and shows as   Christianity  [bible] would go to http://biblegateway.com and shows as   Currently, they are all the same, for all sites. I would like devs (mods maybe) to be able to add them on a per site basis. This is just a selection of examples. People can comment or answer with ones they would like and I'll edit them into this question. I asked this because  on Ask Ubuntu Meta wasn't possible right now - because it wasn't a feature.  [The devs] said right now it isn't possible because magic links are network wide, and they weren't sure it was high enough priority to really get done anytime soon, but if you want you can post a feature-request on Meta Stack Exchange requesting site specific magic links and see if you can garner more support for the feature and maybe get a dev to champion it.  So that's what I've done! Does IBIS reduce image resolution? How does it compare to lens based IS?