Let $f$ be a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the derivative $f'(x)$ and all its successive derivatives exist everywhere and also $f(x)f''(x)\leq 0$ for all $x\in \mathbb{R}$. Let $\alpha$ and $\beta$ be two consecutive roots of $f(x)=0$. Prove that $f'''(\gamma)=0$ for some $\gamma \in(\alpha,\beta)$  I feel that the problem may not be true  I have been able to draw the curve of such a function..
Let $f:R\rightarrow R$ be a function such that all its successive derivatives exist in all $R$ and also $f(x)f''(x)\leq 0$ everywhere.  If $\alpha$ and $\beta$ be two successive roots of $f(x)=0$.   Then prove that $f'''(x)=0$ for atleast one $\gamma \in(\alpha,\beta)$.  My Attempt:  I began by taking an example $f(x)=(x-1)(2-x)$ and let $x=1$ and $x=2$ be its two successive roots.The statement is trivially true.  Then I took the function $f(x)=e^{-x}(x-1)(2-x)$.The statement is true here also so on and so forth.  But what would be the exact proof I wonder. It appears to be a question of Rolle's Theorem.
Lie bracket for Lie algebra of $SO(n,m)$ How does one show that the bracket of elements in the Lie algebra of $SO(n,m)$ is given by  $$[J_{ab},J_{cd}]  ~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$  where $\eta$ has is the definite symmetric form with signature $(n,m)$? Navigate folder structure in Open/Save File Dialog with keyboard? JQuery to load Javascript file dynamically What is a token? Please forgive such a basic question, but I did some Google searching and looked around on this site,  and while there are many threads and online resources that involve tokens, I can't seem to find a direct explanation of what a token is, specifically in the context of TeX, LaTeX, etc. Perhaps there is a good explanation of tokens in the TeXbook, but I don't have a copy of it.   gives me some idea:     A token is a string of characters, categorized according to the rules as a symbol (e.g., IDENTIFIER, NUMBER, COMMA).   but this is a bit too general for me to understand. Taking a guess based on the Wikipedia definition and , is it correct to say that a token is "a thing that has a catcode"? (which, even if true, does not seem entirely satisfactory as an answer: what characterizes the things that are given catcodes?)  Some illustrative examples, and non-examples, of tokens in the realm of TeX, LaTeX, etc. would be much appreciated. that looks an interesting book; they look good guys That looks an interesting book. ()   ‘Look’ can take an NP for its complement by OALD. Now what I’m wondering about is if a person/people could be a subject in the structure, for instance, “They look good guys.” If we can’t, does it have to be “They look like good guys”? I'd like to remove my phone number from my iMessage account, but still have my phone enabled to receive messages sent to my email address (also used on iPad, Mac). Is there a way of doing this? How do I format a Microsoft JSON date? I'm taking my first crack at  with jQuery. I'm getting my data onto my page, but I'm having some trouble with the JSON data that is returned for Date data types. Basically, I'm getting a string back that looks like this:  /Date(1224043200000)/   From someone totally new to JSON - How do I format this to a short date format? Should this be handled somewhere in the jQuery code? I've tried the jQuery.UI.datepicker plugin using $.datepicker.formatDate() without any success.  FYI: Here's the solution I came up with using a combination of the answers here:  function getMismatch(id) {   $.getJSON("Main.aspx?Callback=GetMismatch",     { MismatchId: id },      function (result) {       $("#AuthMerchId").text(result.AuthorizationMerchantId);       $("#SttlMerchId").text(result.SettlementMerchantId);       $("#CreateDate").text(formatJSONDate(Date(result.AppendDts)));       $("#ExpireDate").text(formatJSONDate(Date(result.ExpiresDts)));       $("#LastUpdate").text(formatJSONDate(Date(result.LastUpdateDts)));       $("#LastUpdatedBy").text(result.LastUpdateNt);       $("#ProcessIn").text(result.ProcessIn);     }   );    return false; }  function formatJSONDate(jsonDate) {   var newDate = dateFormat(jsonDate, "mm/dd/yyyy");   return newDate; }   This solution got my object from the callback method and displayed the dates on the page properly using the date format library. Find $\lim_{n\to \infty}({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}})$ strong textFind $\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)$. I do know it is bounded by $1$. I tried using the sandwich rule with no success. How can I solve it? To start in or at a company? I am not sure about what is right:     I can start my career at an international company.   or     I can start my career in an international company.   For me at sounds appropriate? Analysing the "kind(s) of problem(s) are [ones]/[to be]" construction A. These kinds of problem are to be avoided. B. These kind of problems are to be avoided.  Are both A and B grammatical? If not, why not? If so, after having noticed the parallelism between A and C, and between B and D, could we say that C and D are &quot;standard&quot; English, too?  C. These kinds of problem are ones to avoid. D. These kind of problems are ones to avoid. Increase partition size on which Ubuntu is installed? I have Ubuntu installed on a drive. The problem is it is running low on disk space. This is installed on vmware. I expanded the virtual drive and booted into ubuntu. But when I opened gparted(sudo gparted), the move/resize option is unavailable. This is the partition Ubuntu is installed on, but I need to resize it. Any ideas? I am comfortable using command line A $60\times 60$ board is partitioned into rectangles of size $2\times 5$ (or $5\times 2$). Is it true that there always exist another partition into rectangles of size $1\times 3$ (or $3\times 1$) such that any $2\times 5$ (or $5\times 2$) rectangle contains a $1\times 3$ (or $3\times 1)$ rectangle?  For the "simple" partition into $2\times 5$ rectangles, this is certainly true: use the simple partition into $1\times 3$ rectangle.  Another way to partition is to first partition the $60\times 60$ board into $12\times 10$ boards, and then for each such board, put two $2\times 5$ at the top and ten $5\times 2$ below. It is not hard to tile each such $12\times 10$ board with $1\times 3$ (or $3\times 1)$ rectangles so that each $2\times 5$ (or $5\times 2$) contains a $1\times 3$ (or $3\times 1)$ rectangle.  (I'm assuming that an $m\times n$ rectangle has $m$ rows and $n$ columns.) I asked a question regarding a very specific feature of a program and whether or not it works with a larger program.  Within hours someone had marked it with "This question may already have an answer here of a similar question, but not related to my question.  The question I asked IS NOT ADDRESSED in the referred-to question. It's frustrating to spend half an hour putting together an intelligent and specific question, and then have someone mark it as a duplicate (or possible duplicate), which apparently causes people to skip over my question. I haven't received any more responses since the question was flagged.   Two questions:  (1) How do I get this flagging removed and (2) what's the best way to prevent these "helpful" flaggings in the future? Shortest code to calculate list min/max in .NET I'd like something like  int minIndex = list.FindMin(delegate (MyClass a, MyClass b) {returns a.CompareTo(b);});   Is there a builtin way to do this in .NET? I studied the theoretical part about the Gravitational waves in General Relativity (linearization of gravity and small perturbations of the metric and so on).  But I was wondering about: since electromagnetic radiation is composed/carried by Photons (or better:the EM force), shall gravitational waves be composed/carried by Gravitons?   In the end:do gravitational waves imply the existence of gravitons? Or it's something unrelated and off topic? Determine if Term has Grandparent/Great-Grandparent Why does inertia happen? In  R P Feynman relates a story where his father told him that, even though we know the word "inertia" and what it means, nobody knows why inertia happens. Is that still true? Groups of order $pq$ without using Sylow theorems If $|G| = pq$,  $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems? Saving high-quality graphics through commands I am trying to run a function from  in ArcMap 10.1; Sink evaluation. But before it can finish it returns an error message, that, frankly, I don't understand at all.  It goes System.runtime.interopservices.comexception (0x80004005): error hresult e_fail .  Can anyone help me solve this? I have tried all I can think of. Run the function from the toolbox instead of dropdown menus, shorten paths, restart ArcMap, reinstall AHT, open new map documents, restart my computer, wait till next day and see if it automagically have fixed itself...   After searching this forum I found a , although it's got no answers. Now I'm all out of ideas, can anyone please help me?!    I'm using ArcMap 10.1 SP1 basic license and Windows 7.    I think I have narrowed it down a bit, still not clear what's wrong though. Sink evaluation uses a series of functions in Arcmap, of which one is Zonal statistics as table. That's the one it gets stuck on. The input raster somehow does not exist, path unknown. I guess this is some intermediary raster.       After consulting the Arc Hydro team, I got the advice not to use grids stored in a gdb (which I didn't). I have also tried to change target locations, thanks to @G.Oxsen. Unfortunately neither of those solved the issue. According to the Arc Hydro team, GPL0 that I asked about earlier is an in-memory grid, so probably no problem there.   I am currently pursuing that it is related to the installation of SP1. Anyone out there using AHT with SP1 and got Sink evaluation working?    Uninstalling SP1 did not change anything, unfortunately. Still working on this, but with less intensity. Seems like I'll have to find workarounds...    By reducing the number of sinks in need of processing I managed to get Sink evaluation running. Having the files not stored in a gdb seem to help as well, as mentioned earlier, and using integer grids.  Changing the "Maximum number of unique values to render" in ArcMap Options (Customize menu), tab Raster --> Raster dataset is another tip that I got (from the Arc Hydro Team). Restarting ArcMap after running every one or two functions also helps keeping things stable.