Let $I \subseteq \mathbb R$ be an interval and $f:I \to \mathbb R$ increasing and convex. How can  I show that $f$ is bounded from below by an affine function, i.e. there exist $a \in \mathbb R$ and $b \in \mathbb R_+$ such that $a+bx \leq f(x)$ for all $x \in I$? What similiar statement would we have for concave function?
A function $f : I \rightarrow \mathbb R$ on an interval $I$ is convex   if $f((1-t)x+ty)\le (1-t)f(x)+tf(y), \forall x,y \in I, t \in [0,1]$      Assume now that $I$ is an open interval.  Show that if $f$ is convex   then for each $c \in I$ there exists $m \in \mathbb R$ such that $m(x  − c) + f(c) \le f(x)$ for all $x \in I$ , and if in addition $f$ is   differentiable at $c$ then $f'(c)$ is the unique $m$ that works. In   general, must m be unique?   (we have previously shown that convex functions are continuous)  I think I'm missing the point of the question.  Surely, by the mean value theorem we can find some $y\in (x,c)$ such that $f'(y)=\frac{f(x)-f(c)}{x-c}$ setting $m=f'(y)$ we get $m=\frac{f(x)-f(c)}{x-c}, m(x-c)+f(c)=f(x)$ such an $m$ fits the inequality we're asked to show.  I also cannot see why $f'(c)$ would be the unique solution or why we would ever get a unique solution of the inequality at all for that matter.  Clearly I'm misunderstanding the question, so if someone could point out where I would appreciate that.  Thank you
When is it appropriate to use the original pronunciation of a foreign word versus the English pronunciation? When reading to an audience, or speaking in conversation, when is it appropriate to use the original pronunciation of a foreign word versus the English pronunciation (assuming you know the appropriate pronunciation for it)? Is it considered rude, or condescending? Or is it considered a mark of being knowledgeable?  One of the things to consider are place names. When referring to Paris, France, should it be pronounced with the silent 'S' as the French would say it, or with the 'S'? Should Hiroshima be pronounced as a Japanese speaker would pronounce it slightly more emphasis on RO, rather than the SHI? Should Mount Pinatubo be pronounced as a Filipino speaker would pronounce it with shorter stronger vowels, or the longer vowels? (e.g. Pi as is 'pick' rather than 'pea'.) Should Wichita be pronounces as the original "shi" rather than the modern "chi"?  Pronounce pesos or sombrero as a Spanish speaker 'eh' or the English 'ay'?  (I can't think of other common words right now that aren't words taken from other languages like hurricane, boondocks, tornado, etc. which I think have (correctly) changed to English pronunciations.) How do I synchronise two folders in real-time in Windows 7? Integration of $\int\frac{1}{x^{4}+1}\mathrm dx$ How to debug JavaScript / jQuery event bindings with Firebug or similar tools? The successive difference of powers of integers leads to factorial of that power. Here's the formula:     $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$   Can anyone give a proof of this result?  Note: The original question was to prove the more general     $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$$   for an integer $l$, which some of the answers address. Display only relevant reputation points How many "polygons" does my model have? I have my model and in the info bar its show this   May I need to know how many polygons it has, but  there is no option to find polygons. Verts, Edge and Faces nothing polygon option there. In the website its say that the upload limit is 400,000 polygons. I did not get over that limit or how many polygons my model has? I am new to this and not sure what to make of all the options available. I am interested in putting a machine-learning-related preprint, which will potentially be submitted to places like NIPS/ICLR/ICML in the future.   arXiv.org perpetual, non-exclusive license to distribute this article (Minimal rights required by arXiv.org)  Creative Commons Attribution license (CC BY 4.0)  Creative Commons Attribution-ShareAlike license (CC BY-SA 4.0)  Creative Commons Attribution-Noncommercial-ShareAlike license (CC BY-NC-SA 4.0)  Creative Commons Public Domain Declaration (CC0 1.0) Allow me to get notifications for new Meta posts, like mods do I'm currently helping out on . One of the most valuable things on new beta sites, especially private/early public betas, is Meta participation - see :  Treat Meta Questions Like Gold It’s not unusual for your meta site to receive significantly less traffic than the main site. The problems begin when you meta starts to get ignored by the community and ignored by the moderators. It is particularly disturbing when users reach out in meta and starts talking about improving the site or asking for help… only to be ignored; No follow-up, only deafening silence  Because of this, Diamond mods get a notification in their inbox for each new question on their Meta, helping ensure that each meta post is addressed. However, wouldn't it be great if Diamond mods weren't the only ones that could do this? The post does say that the community needs to be seeing these questions too. As an involved user, I would really like to be notified of these new questions - it would increase the good I could do the site. How about this?:  Of course, it would be opt-in, not opt-out, and per-site. A feature like this would greatly improve the good that enthusiastic site-builders can do for the site, and would, most likely, increase the overall chances of a site's success. Travelling within the US as a foreign citizen without a passport I'm travelling within the US and am a British citizen. Can I travel without my passport and just show my UK driver's license instead? Is "mens" a valid word? I've been living in Ireland for almost a year now and I start noticing they use the word "mens" a lot. I can see it used in:   Shops, to denote the area where you can find men's clothes In sport, when they talk about "mens team".    My guess is just that they are lazy about the use of quotes, so that mens should actually be men's. However, there may be some rule I'm not aware of. So, is "mens" only limited to Irish English? When I'm allowed to use it? How do I install the latest version of node.js? How do I install the latest  on Ubuntu? I've been looking around, and I can't find anything. Is there a Ubuntu package for node.js, or do I have to compile it myself? This is the first part of the first half of the a proof in the Axiom of Choice. The proof that the Axiom of Choice is equivalent to the choice function. REsources: Keith Devlin's Joy of Sets: pg. 58. F.R. Drake and Singh Intermediate Set Theory: pg. 110).   I am confused on proving disjointness and will ask further questions of proof once I get this first part v. clear for myself.   AC1: Let $\mathcal F$ be a family of pairwise disjoint, non-empty sets. Then there is a set $\mathcal M$ that consists of precised one element from each member of $\mathcal F$  AC2: For every family $\mathcal F$ of non-empty sets, there is a choice function $\mathcal f$. (I.e. $\mathcal f(X)$ is a member of X, for each X in $\mathcal F$.   Disjointness: Two sets are disjoin if Q $\cap$ R = $\emptyset$. Meaning that they have zero elements in common. These sets do not overlap.   Show: AC1 $\leftrightarrow$ AC2   (1) ($\rightarrow$) Let $\mathcal F$ be a set of nonempty sets.   (2)For each X $\in$ $\mathcal F$, let $X^*$ = (X $\times$ {X}) = {(x, X) | x $\in$ X})  (3) Show Disjointness.  An outline of how book proves  disjointness:  (a) if u = {(X $\times$ {X}) | x $\in$ X} then u is of the form $\lt  z, t \gt$ where z = X and t=x, i.e. x $\in$ X.   (b) then if u $\in$ ({X} $\times$ x) $\cap$ ({Y} $\times$ Y) $\rightarrow$ u = $\lt z, t \gt$ with z = X and z=Y, i.e. X=Y. Which means ({X} $\times$ x) = ({Y} $\times$ y).  Questions: (1) Apparently we cannot prove disjointness in $\mathcal F$ but must prove in a different set. If so, why is that?  (2) why is the case that if u $\in$ ({X} $\times$ X) $\cap$ ({Y} $\times$ Y) then z =X and z=Y? I am not following that step at all.   (3) why does the axiom of choice have to do with ordered pairs? Why are we setting X $\in$ $\mathcal F$, let $X^*$ = (X $\times$ {X})?    Thank you! Views indicator inaccurate, shows one views on question with two close votes Two close votes and only one person has viewed the question?    How did this happen? Ought we only form beliefs based on sufficient empirical evidence? $f$ has the form $f(x) =ax^2+bx+c$. Every differentiable function $f:R \rightarrow R$ with the property that $(2h)f′(x) =f(x+h)−f(x−h)$ for all $x \in R$ and all $h$ has the form $f(x) =ax^2+bx+c$.  I would like to get some help on this one. Taking partial derivatives was an idea, but is that legal? How is the system call in Linux implemented? When I invoke a system call in user mode,how did the call get processed in OS?  Does it invoke some some executable binary or some standard library?  If yes,what kind of thing it needs to complete the call? The Wiki page on  says     The pair $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$.   This is obviously connected to Wilson's Theorem. Can anybody provide a proof for that, along the lines of thought given here, ? Three files have suddenly appeared in my home directory, called "client_state.xml", "lockfile", and "time_stats_log". The last two are empty. I'm wondering how they got there. It's not the first time it has happened, but the last time was weeks ago; I deleted the files and nothing broke or complained. I haven't been able to think of what I was doing at the time reported by stat $filename. Is there any way I can find out where they came from?  Alternatively, is there a way to monitor the home directory (but not sub-directories) for the creation of files? leaving oil on a frying pan for use later I live alone, so I cook for myself. And often times I cook steaks/burgers on a pan, and later in the same day would return to use the same pan/oil without washing/using new oil.  Is this safe, or should one wash and use a new pan before cooking every steak/burger patties? I've actually done this for more than a day once (cooked burgers in the morning, steaks later that afternoon, and then burgers for the next morning).