Prove that $\sin x + 2x \ge \frac{3x.(x+1)}{\pi} $
Prove that $\sin x+2x\ge\frac{3x(x+1)}{\pi}$ for all $x\in [0,\frac{\pi}{2}]$
How do I choose a macro lens for a Canon APS-C camera? What causes close votes to be annotated with "(deleted)"? How to fix pagination for custom loops? I have added a custom/secondary query to a template file/custom page template; how can I make WordPress use my custom query for pagination, instead of using the main query loop's pagination?  Addendum  I have modified the main loop query via . Why isn't pagination working, and how do I fix it? Plotting a 3d surface in tikz, with a limit to the infinity Output URL of open firefox tabs in terminal Why couldn't Ellie's experience be corroborated by simply re-using the machine? Can photons be detected without being absorbed? I am thinking about a detector that would beep if light passes through it. Is it possible? Calculus of variations -- how does it make sense to vary the position and the velocity independently? which vectors are perpendicular to each other?  $\vec a = (1, -2, 3)$, $\vec b = (5, 4, 1)$, $\vec c = (1, 0, -5)$  Do i just take the dot product of 2 of them. If the dot product they are at $90^\circ$? But how do i know if there perpendicular? Short story set in the near future where all wars are settled on the moon using automated machines? Plot Details/Summary  Another one where I have only sketchy memories of the tale, I fear.   The setting for this one is the near future (from the writer's perspective, which I think was the 1970s).  Man's reached a point where bloodshed and loss of life are no longer necessary to settle conflicts.  Instead, all disputes are settled through an annual combat held on the surface of the moon, using the best combat machines each nation can come up with.  The protagonist, as I recall, is an engineer for the Americans.  He recollects past conflicts, and the circumstances for this years fracas.  Much of the combat is told from his observers' perspective.  The actual conflict is a rather clever affair, with a description of each team's designs and all the tricks and gee-whiz things they built into them.  The protagonist's team manages to prevail, when they managed to use an attack as a feint, with the real attack being a swarm of microscopic machines (they did not use the term nanites) that destroyed the opposing machine from the inside.  Timeframe/Other Details  I think this one dates from the 1970s.  I don't think it more recent, given my recollections of when I read it, and also the technology descriptions they used in it.  There is an outside chance that the combat is not between rival nations, but rival corporations.  That would change the premise of "man evolving beyond bloodshed", of course.  I think it's nations fighting, though.  I am almost 100% certain this was in a sci-fi magazine, not an anthology. I recently re-watched Contact for the first time since my childhood, and while I thoroughly enjoyed it, I realized that there was something I'd never understood: why couldn't they corroborate Ellie's account of her journey?  As far as I could tell, the pod fell harmlessly through the machine.  That is, it didn't break anything, and the machine wasn't destroyed.  I don't recall any dialogue about circuits being fried or the system shutting down, so presumably it wouldn't be hard to simply hoist the pod up and go again.  I imagine the energy draw was substantial on the local Japanese power grid, but given the amount of effort and money already invested in the machine, surely "Round 2" wasn't out of the question?  I mean, they almost certainly expected that this machine would get more than one good use out of it, anyway!  So given Ellie's vehement insistence that the machine had worked, the public nature of the controversy surrounding her testimony, and the government's (secret) findings that her device had recorded 18 hours of static, why wasn't it the obvious choice to have someone else suiting up the next day, ready to drop through the machine and corroborate her story? I use Microsoft Outlook 2013 for work and Google Calendar for everything else. I'd like to find an easy way to sync my work calendar to my personal calendar. Ideally, this would be a software application that sits in the background and automates the syncing.  Requirements   Syncing should be one-way only: Outlook to Google. Must support repeating calendar items. Must allow me to specify which calendar in Google to sync to. I will set up a separate calendar for work items. No intermediate server. Export is direct.   Nice to Haves   Automated. Syncs in real time or regularly (say, every fifteen minutes or so). "I'm not sure what the right way is" I believe the sentence in the title is grammatically correct.   Recently I've seen too many people writing it this way:     I'm not sure what is the right way.   Is it grammatically correct as well? Maybe in a different sense? Or simply misuse by non-native speakers? Idea behind proof that $\frac{21n+4}{14n+3}$ is irreducible for all $n$ I have a proof of $\frac{21n+4}{14n+3}$ being an irreducible fraction for any positive integer $n$. The proof starts with the following: Indeed, from the equality: $$2(21n+4)-3(14n+3)=-1$$ and so on..But how does this equality follow from the fraction? What is the idiom, proverb for "Little problems often become big problems if no one takes the initiative to correct them" What is the idiom, proverb for "Little problems often become big problems if no one takes the initiative to correct them"  Which means in an example that If the employees don’t bother to report a malfunctioning machine or a slip-and-fall hazard, serious injuries could occur – injuries that could have been preventable. Ignoring any hazard can be detrimental to one's business and the safety of their employees. How to render from Eevee in Blender 2.8? I have the Blender 2.8 version from the 21st Sep 2018 I was looking forward to render a test sequence but cannot find a render button anywhere? Does anyone know where to find it? Are there ways to determine if a wall is load bearing? Other than looking up blue prints, which many homeowners may not have, are there ways to determine if a wall is load bearing?  Methods I can think of might include:   Going up in the attic to check if ceiling trusses run perpendicular to the wall If the wall is an exterior wall   Any others? Given the following code (that doesn't work):  while True:     #snip: print out current state     while True:         ok = get_input("Is this ok? (y/n)")         if ok.lower() == "y": break 2 #this doesn't work :(         if ok.lower() == "n": break     #do more processing with menus and stuff   Is there a way to make this work? Or do I have do one check to break out of the input loop, then another, more limited, check in the outside loop to break out all together if the user is satisfied? How can I recover from a "AMD-Action:downloadProduct:SP" during installation of macOS? Just wiped my hard drive on my MacBook Pro from 2013. Went to install a new copy of MacOS Sierra. I get the following error:  AMD-Action:downloadProduct:SP   I looked online and tried change the date using the Utility Terminal and the following command:  date mmddHHMMyy   Then did   csrutil enable csrutil clear   Then I tried clearing the NVRAM using command + option + P + R on startup.  Still no luck. Still getting the same error message. Apple support told me to go to a "computer repair store".  Any advice? With all the various races in the federation, how do the humans manage to dominate it?