If $f\left( x \right) = x\cos \frac{1}{x},x \ge 1$. Then which of the following is the correct option for the domain $x \ge 1$ (A) $f(x+2)-f(x)&lt;2$ (B) $f(x+2)-f(x)&gt;2$ My approach is as follow $T\left( x \right) = f\left( {x + 2} \right) - f\left( x \right) = \left( {x + 2} \right)\cos \frac{1}{{x + 2}} - x\cos \frac{1}{x}$ $T\left( x \right) = x\left( {\cos \frac{1}{{x + 2}} - \cos \frac{1}{x}} \right) + 2\cos \frac{1}{{x + 2}} \Rightarrow T\left( x \right) = x\left( {2\sin \left( {\frac{{\frac{1}{{x + 2}} + \frac{1}{x}}}{2}} \right)\sin \left( {\frac{{\frac{1}{x} - \frac{1}{{x + 2}}}}{2}} \right)} \right) + 2\cos \frac{1}{{x + 2}}$ $T\left( x \right) = x\left( {2\sin \left( {\frac{{2x + 2}}{{2x\left( {x + 2} \right)}}} \right)\sin \left( {\frac{2}{{2x\left( {x + 2} \right)}}} \right)} \right) + 2\cos \frac{1}{{x + 2}} \Rightarrow T\left( x \right) = x\left( {2\sin \left( {\frac{{x + 1}}{{x\left( {x + 2} \right)}}} \right)\sin \left( {\frac{1}{{x\left( {x + 2} \right)}}} \right)} \right) + 2\cos \frac{1}{{x + 2}}$ $T\left( x \right) = 2\left( {\left( x{\sin \left( {\frac{{x + 1}}{{x\left( {x + 2} \right)}}} \right)\sin \left( {\frac{1}{{x\left( {x + 2} \right)}}} \right)} \right) + \cos \frac{1}{{x + 2}}} \right)$ From here onward how do I proceed
I have this function $$f(x) = x\cos(1/x)$$ defined for all $x \geq 1$.      How do I prove that    $$f(x+2)-f(x)&gt;2$$    for all $x\geq 1$?   I found $$f'(x)= \cos(1/x) + \frac{\sin(1/x)}{x}$$ and $$f''(x)=-\frac{\cos(1/x)}{x^3}.$$ How do i use this information?
If $f:U\to V$ is holomorphic and $f'(z)\neq 0$ for all $z\in U$, then$f$ is locally bijective. I am trying to solve the following problem:     Let $f:U\to V$ be a holomorphic function such that $f'(z)\neq 0$ for all $z\in U$. Show that for all $z_0\in U$, there exists a disc $D_\varepsilon(z_0)\subseteq U$ such that $f:D_\varepsilon(z_0)\to f(D_\varepsilon(z_0))$ is bijective.   Atempt:  I am trying to use Rouche's Theorem, but I am stuck on a particular step. Let $z_0\in U$. Since $f'(z_0)\neq 0$, there is a disc $D_r(z_0)\subseteq U$ such that $$f(z)=f(z_0)+(z-z_0)h(z),\quad\forall z\in D_r(z_0)$$ where $h$ is holomorphic on $D_r(z_0)$ and $h(z)\neq 0$ for all $z\in D_r(z_0)$.   Let $0&lt;\varepsilon&lt;r$ to be defined later and fix $w\in D_\varepsilon(z_0)$. To show that $f$ is injective in $D_\varepsilon(z_0)$ is to show that the function $f(z)-f(w)$ has exactly one zero in $D_\varepsilon(z_0)$. But $$F(z):=(z-z_0)h(z)$$ has exactly one zero in $D_\varepsilon(z_0)$, so we might want to apply Rouche's Theorem with $F$ and $$G(z):=f(z)-f(w)-(z-z_0)h(z)=f(z_0)-f(w)=-(w-z_0)h(w)$$ to conclude that $F$ and $F+G$ have the same number of zeros and hence $f$ is injective. But that means that we need to find $0&lt;\varepsilon&lt;r$ such that $|G(z)|&lt;|F(z)|$ on $\partial D_\varepsilon(z_0)$. That is, $$|w-z_0||h(w)|&lt;|z-z_0||h(z)|$$ for all $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$.  Can we find such $\varepsilon&gt;0$?  It seems intuitive since if we expand $h$ in a power series at $z_0$, $h(z)=\sum_{n=0}^\infty a_n(z-z_0)^n$, then $$|w-z_0||h(w)| = |w-z_0|\sum_{n=0}^\infty a_n|w-z_0|^n &lt; |z-z_0|\sum_{n=0}^\infty a_n|z-z_0|^n.$$ for $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$. But this is of course not a proof. Assigning strings to arrays of characters I am a little surprised by the following.  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Given a Java InputStream, how can I determine the current offset in the stream? I'd like something like a generic, re-usable getPosition() method that will tell me the number of bytes read from the starting point of the stream.  Ideally, I would prefer this to work with all InputStreams, so that I don't have to wrap each and every one of them as I get them from disparate sources.  Does such a beast exist?  If not, can anyone recommend an existing implementation of a counting InputStream? Is there a way to make an Oracle query behave like it contains a MySQL limit clause? In MySQL, I can do this: select *  from sometable order by name limit 20,10  to get the 21st to the 30th rows (skip the first 20, give the next 10). The rows are selected after the order by, so it really starts on the 20th name alphabetically. In Oracle, the only thing people mention is the rownum pseudo-column, but it is evaluated before order by, which means this: select *  from sometable where rownum &lt;= 10 order by name  will return a random set of ten rows ordered by name, which is not usually what I want. It also doesn't allow for specifying an offset. Lead in ice block This is a strange question I would like to have some explanation accompanied with:  We have an ice block, with in it a lead block. This entire block is put into a glass of water, resulting in a rise of water level until the edge of the glass. The block stays floating on the surface. What will happen with the water level when the ice melts?  The answer is that it will drop, but how is this even true?? My thoughts are: - block floats on water, so rho(block) &lt; rho(water) - Archimedes law, the Fg of block = Fg of water pushed away. - Since a smaller volume of water has equal mass (and so Fg) to block, and ice is less dense than water, when it melts, no mass is lost, so it will take in less space??? Please help me with this point of thinking. I'm looking for either a single word or phrase that would describe either someone or the action of dismissing someone's opinions as something "socially unpopular", without giving any reasoning why, either just to spite them, because they have a poor counter-argument, have been backed into a corner, or something similar.  For example, someone who supports something politically incorrect (like, say, stronger gun control or help for male domestic abuse) might have their views dismissed as racist or sexist or liberal without actually acknowledging and debating against them, thus labeling the former as something they might not be.  I was talking to a friend earlier about politics, and she complained about Germany's immigration problem. She said one issue was that many politicians that try to support stronger immigration laws would immediately be shot down as racist or a nazi and forced to resign, instead of being presented with counter-arguments. Regardless of whether you agree with them or not, what would describe that?  EDIT: Forgot to specify -- the word or phrase I'm looking for would be used in an informal context. And to address further questions, to be used when the person's opinion has legitimate worth and isn't just "being racist" etc, i.e. the dismisser is incorrect, and is only making these accusations because they can't argue further. (Added emphasis to the last sentence of the first paragraph) Different dependencies for different build profiles Additional needed documents for UK trip if someone will shoulder all expenses Someone (a relative) from the UK (already a naturalized British Citizen) will be financing my trip, as well as taking care of my accommodation. What are the documents I need to submit aside from bank statements? Does the sponsor need to fill-out the Sponsorship Undertaking Form? ()  something to consider from my question: -my relative is a naturalized citizen in the UK. Is proof of naturalization needed? How can I make a div not larger than its contents? In Java, are there clear rules on when to use each of access modifiers, namely the default (package private), public, protected and private, while making class and interface and dealing with inheritance? Can current be induced in a superconductor? Is it posible to change the locale of my device? 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Currently, I redirect all requests to http/80 to https/443, which uses a self signed certificate I created a while ago.  Now - from what I understand Let's Encrypt makes a request to port 80 (as I am using the webroot option of certbot). These requests are redirected, which renders the certificate generation unsuccessful.  I tried to achieve this with the following server block, listening at port 80:  server {         listen  80;              server_name     sub.domain.tld;         server_tokens   off;           location /.well-known {                 root /var/www/letsencrypt;         }          location / {                 return 301 https://$host$request_uri;         } }   But requests to /.well-known are redirected to https/443 anyways.  How can I redirect all requests from http/80 to https/443, except the requests to /.well-known/? Note: I am posting a question from a former student of mine unable to post on his own for technical reasons.  Given an iid sample $x_1,\ldots,x_n$ from a Weibull distribution with pdf $$ f_k(x) = k x^{k-1} e^{-x^k} \quad x&gt;0 $$ is there a useful missing variable representation  $$f_k(x) = \int_\mathcal{Z} g_k(x,z)\,\text{d}z$$and hence an associated EM (expectation-maximisation) algorithm that could be used to find the MLE of $k$, instead of using straightforward numerical optimisation? What does "No bootable medium found" mean in VirtualBox? When should I have an NPC actively investigate an illusion created by the Phantasmal Force spell? The phantasmal force spell can be broken if the victim examines the illusion and makes a successful investigation check against the spell's DC.  What general guidelines should I use to decide when the victim of phantasmal force should use its action to examine the illusion?  Suppose the caster makes an orc think it is being attacked by a wolf.  Should the orc always examine the wolf until the spell ends? Or should the orc use its action to attack the wolf, defend itself, or run away? Why didn't Iron Man make Vibranium armor in Iron Man 2? In the movie Iron Man 2, Tony Stark figured out how to synthesize Vibranium. Why didn't he use it to make a new Iron Man armor?