Let $f:[a,b]\to\mathbb{R}$ be differentiable and $f'$ is continuous. Suppose $f(a)=f(b)=0$, show that  $$\max_{x\in[a,b]}|f'(x)|\geq\frac{4}{(b-a)^2}\int_a^b|f(x)|dx$$  My approach. For any $x$, by Taylor's theorem there is $\xi\in(a,x)$ s.t.  $$|f(x)|=|f(a)+f'(\xi)(x-a)|\leq\max_{x\in[a,b]}|f'(x)|(x-a)$$  Hence   $$\int_a^b |f(x)|dx\leq\max_{x\in[a,b]}|f'(x)|\int_a^bx-adx=\frac{(b-a)^2}{2}\max_{x\in[a,b]}|f'(x)|$$  I can only get this, and I don't know how to obtain $\frac{(b-a)^2}{4}$.
Let ${\rm f}:\left[a, b\right]\to\mathbb{R}$ be twice differentiable, and suppose   $$\lim_{x\to a^{+}} \frac{{\rm f}\left(x\right) - {\rm f}\left(a\right)}{x - a} = \lim_{x\to b^{-}}\frac{{\rm f}\left(x\right) - {\rm f}\left(b\right)}{x - b} =0 $$  Show that there exists $\xi \in \left(a, b\right)$ such that $\displaystyle{% \left\vert\vphantom{\Large A}\,{\rm f}''\left(\xi\right)\right\vert \geq \frac{4\left\vert\vphantom{\Large A}% \,{\rm f}\left(a\right) - {\rm f}\left(b\right)\right\vert} {\left(b - a\right)^{2}}}$.  I don't know how to start. Any hints ?.
How can I find out the variable name in python? I know how I can find out the python command by clicking a button in the Blender interface, and then look in the info window:   For example, as can be seen, I added a cube, UV sphere, plane, camera, lamp, pressed A twice.  But how can I find out an object variable name, for example the y-rotation of an object. Because I can't find the y-rotation, when I point my mouse over it, I can only find the rotation: Center of a group ring? Let $R$ be a ring with unity and let $G$ be a group. Then what is the center of the group ring $R(G)$? I feel that the center $Z(R(G))$ is $R(Z(G))$. I assumed if $x=\sum_{i=1}^{n} r_i g_i$, then it should commute with every element of $R(G)$. In particular it should commute with $g=1 \cdot g \in R(G)$ Thus we get $$\big(\sum_{i=1}^{n} r_i g_i\big)g=g\big(\sum_{i=1}^{n} r_i g_i\big)$$ Thus I get $$g_i \cdot g=g_i \cdot g \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \forall g \in G$$ Hence $x \in Z(R(G)) $. Is this correct? How to make an image center (vertically & horizontally) inside a bigger div I have a div 200 x 200 px. I want to place a 50 x 50 px image right in the middle of the div.   How can it be done?  I am able to get it centered horizontally by using text-align: center for the div. But vertical alignment is the issue.. "Never say die?" What's the deal with this idiom? I know what it means, but cannot wrap my head around its grammar structure. The sentence structure of     "Never say die"   looks like that of     "Never admit defeat"   but it does not make the same grammatical+semantical sense.  Is it an archaic use? Is it a contraction of a longer phrase? Was it originally misspoken by someone of note and survived memetically? What are its roots?  Edit: Well, the comments made me realize that I worded my question poorly. I'm not saying I think this phrase is ungrammatical. I'm asking how the heck "saying 'die'" came to mean what it means. Other users have been leaving me angry and hostile comments accusing me of plagiarism and saying that I'm a liar and I should stop stealing answers. But I have only been copying content from other sites, not taking answers from other users.   I don't understand what is wrong here. Shouldn't we be focused on building all the content and knowledge in a single place, much like Wikipedia does?  What does it matter that I copy and paste content I've found elsewhere so long as the question gets an answer? The glorified global variable - becomes a gloried global class. Some say breaking object-oriented design.  Give me scenarios, other than the good old logger where it makes sense to use the singleton. How to install Oracle 18c (Enterprise Edition) on Ubuntu 18.04? This question is unanswered since Oracle 18c has been released. Except Ubuntu Linux platform, most of the other Linux platform supports Oracle 18c installation. I, myself, tested this on CentOS 7 and oracle 18c got installed without any glitch!  It is unfortunate that Oracle Corporation has not documented the installation procedure particularly for Ubuntu Linux platform, nevertheless one good thing is that the image file db_home.zip is a generic one, installable across all Linux platforms.    So how can I install Oracle 18c (Enterprise Edition) on Ubuntu 18.04? Select Permissions Across Multiple Schemas Within the Same Database I have been asked by a client to implement some views for reporting purposes, the views will be accessed via PowerBI, Excel and SSRS. The designated user will only have access to the view and no underlying tables must be available to the user.  The problem I have encountered in that the SQL within the view touches 3 different schemas (All within the same database):   Pupil Provider  Security   The view is:  CREATE VIEW dbo.vTestPermissions AS SELECT a.Column1,        b.Column1,        c.Column1 FROM Pupil.Table1 a JOIN Provider.Table2 b ON a.Column1 = b.Column1 JOIN Security.Table3 c ON a.Column1 = c.Column1   The owners of the Tables/View are as follows:   Pupil.Table1 - Owner Pupil Provider.Table2 - Owner Provider Security.Table3 - Owner Security vTestPermissions - Owner dbo   When I select from the view I get an error:     The SELECT permission was denied on the object 'table3', database   'TEST', schema 'Security'   I have tried giving SELECT permission with and without the GRANT option to the schema, and the tables, but that makes the underlying tables available to the user.  Any help on this will be greatly appreciated. Is this functionality going to be put into a later Java version?  Can someone explain why I can't do this, as in, the technical way Java's switch statement works? What is syntax highlighting and how does it work? I noticed that sometimes my code gets highlighted in different colors when rendered.   What is syntax highlighting? How does it work? Why isn't my code being highlighted correctly? How do I report a bug or request a new language? How do I use syntax highlighting? What languages are currently available on Stack Exchange? Is it possible to recover Classical Mechanics from Schrödinger's equation? Let me explain in details. Let $\Psi=\Psi(x,t)$ be the wave function of a particle moving in a unidimensional space. Is there a way of writing $\Psi(x,t)$ so that $|\Psi(x,t)|^2$ represents the probability density of finding a particle in classical mechanics (using a Dirac delta function, perhaps)? Robot tank run amok, its trainer trapped in cave on the Moon Another dim memory from my sci-fi reading binges in the 1980s... this one almost definitely in one of the big hardback collections full of late Golden Age short stories.  The story is told from the perspective of a robot (maybe cyborg?) tank that has stopped responding to control signals and is aggressively "defending" some portion of the Moon, to which it was previously assigned. Maybe a war had ended, or maybe it became confused about an exercise that it thought was real?  There was a person trapped in a cave trying to communicate with the tank. The tank mostly ignores him, so the main purpose of what he says is exposition for the reader. I think this person was an expert on this type of robot/cyborg sent to "talk it down", but the tank had attacked, and this guy was the only survivor. The expert was badly wounded -- I think he mentions his suit had to amputate his leg to seal a hole -- or maybe flood it with water which would freeze to seal the hole but mean that his leg had to be amputated later. He keeps trying to convince the tank to stand down, at one point saying that he knew it "back when it was just a two-wheeler an autocyber in boot camp" or something along that line. (Edit: I must have been reading too fast and/or distracted back then, since per the quote below I seem to have misread "motorcycle" for "autocyber"!)  One detail of the attack stands out -- the tank was low on ammunition, I think, so it used a kind of bomb with an electric/magnetic "catapult" (aka "magnapult") to hit the expert's transport. The tank was proud of itself for figuring out a way to hit it far out of the normal range by 1) using its own top speed to add velocity to the projectile, and 2) building up the catapult's energy past its normal operating range, which caused it pain but made the bomb go much further. It catches the transport by surprise, since they thought they were safely out of range of the armaments it had left.  The premise sounds a lot like a Keith Laumer "Bolo" story, but I'm fairly sure that it's not actually one of them. It had a somewhat grimmer feel to it. I could be wrong, though, since it's been 30 years. My settlement was attacked by some Raiders and my defenses took care of them along with some of my NPC's. It seems like the Raiders didn't make it through to any of the buildings. What happens when they do?  Can you lose your items if you stored them in a house?  Can the NPC's die?  Do the enemies destroy your buildings? Python is quite cool, but unfortunately, its debugger is not as good as perl -d.    One thing that I do very commonly when experimenting with code is to call a function from within the debugger, and step into that function, like so:  # NOTE THAT THIS PROGRAM EXITS IMMEDIATELY WITHOUT CALLING FOO() ~&gt; cat -n /tmp/show_perl.pl 1  #!/usr/local/bin/perl 2 3  sub foo { 4      print "hi\n"; 5      print "bye\n"; 6  } 7 8  exit 0;  ~&gt; perl -d /tmp/show_perl.pl  Loading DB routines from perl5db.pl version 1.28 Editor support available.  Enter h or `h h' for help, or `man perldebug' for more help.  main::(/tmp/show_perl.pl:8):    exit 0;  # MAGIC HAPPENS HERE -- I AM STEPPING INTO A FUNCTION THAT I AM CALLING INTERACTIVELY   DB&lt;1&gt; s foo()  main::((eval 6)[/usr/local/lib/perl5/5.8.6/perl5db.pl:628]:3): 3:      foo();     DB&lt;&lt;2&gt;&gt; s main::foo(/tmp/show_perl.pl:4):     print "hi\n";     DB&lt;&lt;2&gt;&gt; n hi main::foo(/tmp/show_perl.pl:5):     print "bye\n";     DB&lt;&lt;2&gt;&gt; n bye     DB&lt;2&gt; n Debugged program terminated.  Use q to quit or R to restart,   use O inhibit_exit to avoid stopping after program termination,   h q, h R or h O to get additional info.      DB&lt;2&gt; q   This is incredibly useful when trying to step through a function's handling of various different inputs to figure out why it fails.  However, it does not seem to work in either pdb or pydb (I'd show an equivalent python example to the one above but it results in a large exception stack dump).  So my question is twofold:   Am I missing something? Is there a python debugger that would indeed let me do this?   Obviously I could put the calls in the code myself, but I love working interactively, eg. not having to start from scratch when I want to try calling with a slightly different set of arguments. I have to format my entire disk and reinstall Ubuntu.  I have installed a lot of software on my current system. And I will have to reinstall all those updates, drivers and applications too.  Is there any way I can make an .iso image of my installed Ubuntu along with all the software, preferences, drivers etc in it ?  Is there any way I can back up my whole OS along with all its software, drives, updates and all ? Prove that if $n$ is not the square of a natural number, then $\sqrt{n}$ is irrational. Integral representation of Euler's constant Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$  where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$).    This integral was mentioned in  as in  , but the solutions I've got uses corollaries from . Can you give me a simple solution (not using much advanced theorems) or at least some hints. Convergence in measure - product I'm trying to prove the following statements in Folland's book.  Let $(X,\mathcal{M},\mu)$ be a measure space. If $f_n\to f$ in measure and $g_n\to g$ in measure, then $f_n+g_n\to f+g$ in measure and $f_ng_n\to fg$ in measure (in this last case, assume $\mu(X)&lt;\infty$).  The first one is already done, but the second one is giving me trouble.   My attempt: We can take subsequences $f_{n_k}$ and $g_{n_k}$ such that $f_{n_k}\to f$ a.e. and $g_{n_k}\to g$ a.e. In this case, we know that $f_{n_k}g_{n_k}\to fg$ a.e. Note that $\mu(X)&lt;\infty$, so we can use Egoroff's theorem to conclude that $f_{n_k}g_{n_k}\to fg$ almost uniformly. Finally, almost uniform convergence implies convergence in measure, therefore $f_{n_k}g_{n_k}\to fg$ in measure.   The problem is that I just have convergence in measure for some subsequence of $f_ng_n$ and I need convergence in measure for the whole sequence. Also I don't know if using Egoroff's theorem is unnecessary, if this is the case, I would appreciate to see some more elementary prove.  Detail: We say that $f_n\to f$ almost uniformly if for every $\varepsilon &gt; 0$ there is some $E\subset X$ measurable, such that $\mu(E) &lt;\varepsilon$ and $f_n\to f$ uniformly on $E^c$.  Thank you! List has no rows for assignment to SObject How can I access google.com outside the US? I live in the UK, and whenever I visit google.com I am redirected to google.co.uk, but I want to see my website's rankings on the US version (I presume the results are different). Is there a way to do this?