If $a&gt;b&gt;0$ and $n\in\mathbb{N}$ then $a^{1/n}-b^{1/n}&lt;(a-b)^{1/n}$  I know $f(x)=x^{1/n}-(x-1)^{1/n}$ is strictly decreasing if $x&gt;1$. Since after taking the derivative I get that $f'(x)=\frac{1}{n}x^{\frac{1-n}{n}}-\frac{1}{n}(x-1)^{\frac{1-n}{n}}$ which is decreasing for all $x&gt;1$  Now evaluating $f(1)$ I get just $1$ and evaluating $f\left(\frac{a}{b}\right)$ I get $\frac{a}{b}^{\frac{1}{n}}-\left(\frac{a-b}{b}\right)^{\frac{1}{n}}$. Now I'm not sure how to then show that  $a^{1/n}-b^{1/n}&lt;(a-b)^{1/n}$
Let $a&gt;b&gt;0$ and let $n \in N$ satisfy $n \geq 2$.  Prove that $$a^{1/n}-b^{1/n}&lt;(a-b)^{1/n}.$$  If we let $a=x$ and $b=x-1$, then we need to show that $f(x):=x^{1/n}-(x-1)^{1/n}$ is decreasing for $1\leq x$ (which means that $f'(x)\leq 0$ and $f$ is differentiable), and evaluate $f$ at $1$ and $a/b$.
Windows Update not working on Windows 2012 R2 Standard I recently inherited the management of a Windows 2012 server at a remote site.    I checked Windows Update and it has not been updating since March.  When I tell Windows to check for updates, it acts like it is checking, but it seems to say that for hours.   If I attempt to restart the windows update service, it seems to never be able to shut down.  My only remedy seems to be rebooting to get back to the point where I can tell Windows Update to check for new updates.    The last successful check for updates says March 20.  The last time updates were installed says March 17 (failed).  The update history shows that one update failed for March 17, a printer driver update, but the history shows 13 failed updates for Feb 17.  Not sure what else to try. What is the official name for the "special" ASP.NET tags like this:  &lt;%# %&gt; &lt;%= %&gt; &lt;%@ %&gt; &lt;%$ %&gt;   I can't seem to figure out the conceptual or well known name for these, so I'm having trouble searching for more info.  As a bonus, can anyone give me a quick rundown of all of the possible "special tags" and what each one of them does (or point me to a resource)? Why are my SVG imports so small? I'm definitely a newbie with blender still, as I've dipped my toe in a bit being a video editor.  My issue is importing .svg saved from illustrator.  Just simple vectors with not too many paths.  Just a thick text company logo.  I want to bring in and extrude, light, etc.  I've done it before in older versions of blender with no problem.  Am I not doing something right? Can someone tell me how to get an eps saved as an svg from illustrator to size correctly? Does Java Connection.close rollback? Why is this clause unsatisfiable? Wild Magic Surge - What does "You cast ..." mean for your spell slots? In response to  question, I started to think about what it means when a Wild Magic Surge results in a "You cast spell-X".  So, "specific beats general" gives:   You can cast a second spell when you would not normally be allowed to You can cast a spell you don't know If the surge forces you to cast a concentration spell you would lose any concentration spell that you currently had active Subject to what the specific entry says you can target it how you like Unless the specific entry prohibits it you could use metamagic on the spell   I am pretty confident in all of the above but if anyone disagrees please address in an answer.  Here comes the question.  How does this impact on your available spell slots?  "You cast ..." to me suggests that if you have an applicable slot available then you have to use it to cast the spell because that is the general rule for casting spells but if you don't then "specific beats general" and it is cast anyway.  Specific examples:     You cast fireball as a 3rd-level spell centered on   yourself   This must be a 3rd level spell:   if you have a 3rd level slot available you have to use it? if you don't, if you have 5 sorcery points and a bonus action available must you use these to get that slot? if you don't have any of that then the spell goes off anyway?      You cast confusion centered on yourself.   No level specified and confusion is a 4th level spell:   if you have a 4th level slot available you have to use it? if you have a 4th, a 6th and an 8th level slot you can choose which one to use? if you have only a 5th level slot but also 6 sorcery points and a bonus action, you could choose to use the 5th level slot or use the points to create a 4th level slot? if you don't have any 4th level or higher slots but you have 6 sorcery points and a bonus action available must you use these to get that slot? if you don't have any of that then the spell goes off anyway?   And so on ... Accidental chown under / as root I accidentally run the following command as root chown mike -Rf / backups the point is that I added a space between / and backups my mistake, I stopped the command after 1-2 seconds, how can I recover from this? I have problems with DirectAdmin right now for example when accessing it I recive the error:  Unable to determine Usertype user.conf needs to be repaired http://help.directadmin.com/item.php?id=456   Or I stopped recieving emails, I dont even know what else is messed up right now... How to fix task manager memory column if it shows lies I have a problem in showing memory in task manager with memory and paging columns. Why it shows so small size usage? And maybe anyone knows how to fix it?  Here is a screenshot(working set, memory, page pool) sorted by most consuming: Interaction with the site on the app should count towards Enthusiast/Fanatic If I do any of the below, which would normally constitute "activity" on any Stack Exchange site (including mobile):   View my profile Vote on any question or comment Visit one or more questions   ...then Stack Exchange considers me to have "visited" the site.  If I do any of the below on the mobile app:   Vote on any question or comment Visit one or more questions   ...then Stack Exchange does not consider me to have visited the site.  If one wants the app to have feature parity with the mobile site, then shouldn't my participation within the app also count towards the Enthusiast/Fanatic badge? The Order of Modification in English Nouns, Preceding or Succeeding? As I don't know the exact linguistic terms, what I mean my "preceding" and "succeeding" in modifying nouns is as follows.  Preceding : delicious food, long way, kind person, et cetera Succeeding : faculty of science, land of despair, a book that changed my mind, et cetera  English seems to accept both ways of modification.  Sometimes, either way will not make a difference. department of science == science department a burning book == a book burning  At other times, changing the order will still make sense but be somewhat awkward or bizarre. Kentucky Fried Chicken ?= Chicken Fried in Kentucky free as in free speech ?= as-in-free-speech free  When writing English, I tend to use more the preceding modification style because my native language Korean is quite strictly precedingly modifying. "빠른 판단을 바탕으로 한 재치있는 응수" directly translates to "a quick-decision-based clever response" rather than "a clever response based on a quick decision"  French as well as other Latin-originated languages seems to be succeedingly modifying. Académie Française, Poulet Frit du Kentucky  while German and Dutch more often use preceding modification.  Well, this mixture in English allows rich style with various ways of expression, but it does often make a non-native English speaker be a little confused. Is there any subtle but general difference between these 2 ways? Or is it just simply case-by-case. Any guidance will help. Validate/verify PDF files integrity When can we use 'much' in affirmative clauses? I found the below in  web-site:     We use much in questions and negative clauses to talk about degrees of something.      We don’t use much in affirmative clauses.   Also, it the same web-site, I found the following:     She runs much faster than he does.    also I noticed that some web-sites use much with affirmative clauses:     May we all be having this much fun and getting this much exercise when we're his age!    Is it Okey to use much in affirmative clauses? If yes, when can we use it? and what is the rules?  Update:  Here is another example on how often the word "much" is used  in informal style:     After much consideration I will NOT be performing at the Inauguration with my improv group Capitol Hill-arious WITH ROLLUP WHERE x IS NULL I've got a PDF that is automatically generated from a certain piece of software, that I want to include in the report I'm writing. I'm using pdfpages which gets the job done elegantly, but the problem is, the PDF has a lot of white space around the actual content (which is a graphics) and I always have to do some tedious post-processing of the automatically generated document so that it fits well in my report. Changing the output of the software to eliminate the white space is not an option (or at least, would be much more trouble than it's worth).  I checked the pdfpages , but wasn't able to find an option that does that. Ideally, I'd like to be able to set margins from each direction, and the PDF is then cropped accordingly and included.  What other options do I have to control what portion of the included PDF is visible in the final document? How to check if a String contains another String in a case insensitive manner in Java? Question mark and colon in JavaScript ADC0808 IC working correctly occasionally I am using  for a project on converting analog signal to digital signal. It has an inbuilt multiplexer using which we have to provide an analog signal to one of the pin. However what happens is ;  Sometimes it latches to a particular value say 1000000 and remains at there no matter how many times Start of Conversion is pressed which should set the inner counter to 0.  Sometimes it gives slightly wrong output off by 100 or 1000 and when I press Start of conversion, it gives correct output, so its not reliable every time.  And most seriously , Suppose the control for multiplexer according to Datasheet are such that input pin 1 should be active, instead, it works only when the input is given to pin 5 for one particular IC and pin 3 for other. I have been trying to fit it in logic since 3 days however each day some new error happens. I have replaced it twice from my college lab and now the technician doesn't want to replace it anymore . What is amiss here ? Topological/Geometrical justification for $\text{CFT}_2$ being special It is known as a fact that conformal maps on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ for $n&gt;2$ are rotations, dilations, translations, and special transformations while conformal maps for $n=2$ are from a much wider class of maps, holomorphic/antiholomorphic maps. I was wondering to know if there is any topological or geometrical description for this.   To show what I mean, consider this example: in $\mathbb{R}^n$ for $n&gt;2$ interchanging particles can only change the wave function to itself or its minus. It is related to the fundamental group of $\mathbb{R}^n-x_0$ ($x_0$ is a point in $\mathbb{R}^n$ and $\pi_1(\mathbb{R}^n-\{x_0\})=e$ for $n&gt;2$) but this is not true for $n=2$.   I want to know whether exists any topological invariant or just any geometrical explanation that is related to the fact that I mentioned about conformal maps on $\mathbb{R}^n$. "Whom" or "who" where the referent is both subject and object? I understand there has been so much on this topic but I am still confused. I get that if the person is the subject it is who and anything else is whom. However, I'm really struggling to work out this rule for the following sentence:     The scope of which his account of morality applies to is therefore restricted to those whom/who the ‘notion of justification to a being of that kind makes sense.’   Should it be who or whom? I can't work out the subject and object. I want to find the probability that mean is less than 5 given that poisson distribution states it is 6 ie find p(x&lt;5|x~po(6))  Here is the actual question: Two grocers agree that the daily demand for a particular item has Poisson distribution. However, grocer A claims that the mean demand is 3 items per day, while grocer B claims that the mean demand is 6 items per day. They agree to resolve the disagreement by observing the demand on one particular day: B agrees to accept A’s claim if the observed demand is 4 or less, and A agrees to accept B’s claim if the observed demand is 5 or more. (a) Calculate the probability that A’s claim is accepted when, in fact, B’s claim is correct