Let  $f:[a,b]\rightarrow \mathbb{R}$ s.t $f(a)=0$. If $f$ is differentiable on $[a,b]$ and there exist $C  \in \mathbb{R}$ s.t $|f'(x)|  \leq C|f(x)|$ for all x $ \in [a,b]$, then $f(x)=0$ for all $x  \in [a,b]$  My attempt:  |$f'(a)| \leq C|f(a)|=0\Rightarrow f'(a) = 0$  Let $\epsilon=\frac{1}{C}$, then there exist $\delta &gt; 0$  s.t for all $x \in (a,b)$ if $x\in (a,a+\delta)$, then $\frac{|f(x)|}{x-a}&lt;\frac{1}{C}$ thus $$|f'(x)|\leq C|f(x)|&lt;|x-a|&lt;\delta$$  I do not know what to do from here.  Edit: I could try to think about $f'(b)$: If $f'(b)&gt;0$ or $f'(b)&lt;0$, then find some contradiction.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and differentiable function in all $\mathbb{R}$. If $f(0)=0$ and $|f'(x)|\leq |f(x)|$ for all  $x\in\mathbb{R}$, then $f\equiv 0$.  I've been trying to prove this using the Mean Value Theorem, but I can't get to the result. Can someone help?
Search by multiple tags I'd like to be able to search for questions by multiple tags; for example "questions tagged 'nhibernate' and 'web-services'".  I can get close by searching for "nhibernate web-services" but the results tend to be a few questions that relate to both terms--but much more largely just questions that relate to one or the other. Move the Add Favorite / Ignored Tags boxes to the top It seems that most people who use the ignored tags feature end up with a fairly long list of those tags.  Although , it is sometimes necessary to add new ignored tags manually (typically when adding some-tag*).  It would be a lot easier if the add box was above the long list of ignored tags rather than below. I suppose the same would apply to the favorites tags for consistency's sake. I might want to answer my own question because:  I arrived at a solution before there were any other correct answers or I like mine better, can I still answer my own question?  I have a folder with lists of code snippets for things I always forget or get wrong, such as regular expressions for processing XML, common WinForms control idioms, etc. Is it considered poor etiquette for Stack Overflow to ask a question I already know the answer to, and then self answer, just so it gets recorded on the website?   Related:          For more information, see &quot;&quot; in the . With reference to:    I removed the Cortana packages from my running Windows 10 using the method posted by magicandre1981 and using  win6x_registry_tweak.   My question is how can I re-install Cortana (or for that matter any other package removed similarly)?  I have opened up the install.wim image (converted from a Win 10 install.esd file) and found Cortana in the system apps folder, but don't know how / what to use to re-install it.  Any help please? Yesterday, one of our computers dropped to grub shell or honestly, I am unsure what shell it was when we turned on the machine.  It showed that it can't mount the root filesystem or something in this sense, because of inconsistencies.  I ran, I believe:  fsck -fy /dev/sda2   Rebooted and the problem was gone.  Here comes the question part:  I already have in her root's crontab:   @reboot /home/ruzena/Development/bash/fs-check.sh    while the script contains:  #!/bin/bash touch /forcefsck   Thinking about it, I don't know, why I created script file for such a short command, but anyways...  Further, in the file:  /etc/default/rcS   I have defined:   FSCKFIX=yes    So I don't get it. How could the situation even arise?    What should I do to force the root filesystem check (and optionally a fix) at boot?  Or are these two things the maximum, that I can do?  OS: Linux Mint 18.x Cinnamon 64-bit.  fstab:  cat /etc/fstab | grep ext4   shows:   UUID=a121371e-eb12-43a0-a5ae-11af58ad09f4    /    ext4    errors=remount-ro    0    1    grub:   fsck.mode=force    was already added to the grub configuration. Storing third-party libraries in source control Should libraries that the application relies on be stored in source control? One part of me says it should and another part say's no. It feels wrong to add a 20mb library that dwarfs the entire app just because you rely on a couple of functions from it (albeit rather heavily). Should you just store the jar/dll or maybe even the distributed zip/tar of the project?  What do other people do? I found an argument online that the two fields are not isomorphic but I can't make sense of the argument. You find it .  These are the things I'm confused about when it comes to the argument.     It says that the field $\mathbb Q[x]/(x^2+1)$ contains a root of the polynomial $x^2+1$. I suppose this is the congruence class $[x]$? But this is not the same as the zero element of the field $\mathbb Q[x]/(x^2+1)$, right?   It says that every isomorfism takes $0$ to $0$. I interpret this as the zero-element in $\mathbb Q[x]/(x^2+1)$ (which I interpret to be $[0]$) is taken to the zero-element in $\mathbb Q[x]/(x^2+2)$ denoted $[0]_2$. But in the argument it says that this means that if $\mathbb Q[x]/(x^2+1)$ has a root of the polynomial, then also $\mathbb Q[x]/(x^2+2)$ must have a root of this same polynomial. Why is that?     Greatful for any help! Prevent questions on Hot List from being upvoted by casual visitors (only rep is from association bonus) ‘It’ – ambiguous antecedent? Is there a way to edit SVG files in Illustrator where you can specify a CSS class for each path?  I already know in Illustrator that if you give the layer name an actual name, Illustrator will use that name as the path's ID, which is fine if you don't plan on reusing the icon more than once on the same page.  My current workaround is to just use the IDs method, but then later convert the IDs to classes in my code editor, but it's pretty annoying to have to do every time I generate my SVG sprite.  If that's currently not possible in Illustrator, are there any other apps that will allow you to specify that? Or maybe a Grunt or Gulp package?  It looks like you might be able to do it with Inkscape with a , so I might look at that if there's no other good solutions out there. Inequality involving sides of a triangle How can one show that for triangles of sides $a,b,c$ that   $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} &lt; 2$$  My proof is long winded, which is why I am posting the problem here.   Step 1: let $a=x+y$, $b=y+z$, $c=x+z$, and let $x+y+z=1$ to get   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}&lt;2$  Step 2: consider the function $f(x)=\frac{1-x}{1+x}$, and note that it is convex on the interval (0,1), so the minimum of   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}$  is reached when the function takes the extreme points. i.e. $x=y=0, z=1$.    But going back to the fact that this is a triangle, we note that $x=y=0 \implies a=0$ which is not possible, so the inequality is strict. If a character who was under the effect of a growth potion (double their height and eight times their weight) and weighed 2,400 lbs. was able to use Dimension Door to teleport 400 feet or more into the air, directly above a Huge-sized monster, and fell on it, how much damage would the falling creature - and the monster - take?  I would assume both would take the 20d6 max for the falling over 200 feet, but is there a estimation on additional damage for the falling creature's size?  According to the splat calculator, at 500 feet you'd be falling at 196 km/hr, and it would expend 1.6 million joules of energy, the equivalent of over 3,000 mid-sized cars hitting an object at 60 km/hr.  A 2,400-lb. barbarian falling 500 feet is like dropping a mid-sized Toyota Corolla off a 50 story building onto a monster - it should do some damage. Why do I need an extended partition? What is an extended partition? Why do I need to use it to install Ubuntu if I already have four partitions that are factory installed? Social Login adoption vs Traditional Login I have implemented  on my website. I am wondering how important it is to include a traditional login (email/password) system.  I believe there may be a significant percentage of potential users that are either uncomfortable or unfamiliar with social login. These users may not sign up for my site if social login is the only option.  Does anyone know of any data comparing social login vs traditional login and the corresponding adoption rate?  Is there a current best practice? Do I need to give the user an option to sign up using username/password? Or is facebook/google/openID/twitter/MS/Yahoo enough? Is there a tool to get the lines in one file that are not in another? How do you compute negative numbers to fractional powers? My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure if the number stays negative or turns into a positive. I know that if a negative number is raised to an odd power it is negative, but fractional powers are neither odd or even. Is there a general rule for dealing with these types of problems? My mother is a quite famous researcher (she has a PhD) on the field I want to do a PhD in. She works on the industry (not academia). I helped her on successful projects. She knows more about my work than my professors since we have worked much longer together.  Is it a good idea to ask for a recommendation letter from her? Since our personalities are very similar she can point out the high probability that I will be successful like her? Prove that $f(x)\in \mathbb{Z}[x]$ such that $f(0)$ and $f(1)$ are odd has no integer roots Will inverse functions, and functions always meet at the line $y=x$? If I have a function, the inverse function, by definition will be a reflection of the original function in the line $y=x$, so if I wanted to find the point of intersection, instead of solving it with equating both of the functions to equaling each other, could I assume that the point of intersection, between the two functions, will always be at the point $y=x$? This would enable me to solve problems much easier, instead of having to solve quartic equations, for example. How to prove those "curious identities"? How to prove $$ \prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}$$ and $$ \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}}$$