How to go upon proving $\frac{x+y}2 \ge \sqrt{xy}$? I'm trying to prove this but am having some difficulty.   For any $x,y\in\mathbb R$ such that $x\ge 0$ and $y\ge 0$ we have $$\frac{x+y}2 \ge \sqrt{xy}.$$    So far what I have gotten to is $\frac{x+y}{2} \geq \frac{\sqrt{xy}}{2} $  After this point I don't know what to do. To get to this point:  $$y \geq 0 $$ $$\implies y \geq y$$ $$\implies xy \geq xy$$ $$\implies 2xy \geq xy$$ $$\implies x^{2} + 2xy + y^{2} \geq xy$$ $$\implies (x+y)^{2} \geq xy$$  Sqrt both sides to get:  $$\implies x+y \geq \sqrt{xy}$$ $$\implies\frac{x+y}{2} \geq \frac{\sqrt{xy}}{2}$$
Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$ I am having trouble with this problem from my latest homework.  Prove the arithmetic-geometric mean inequality. That is, for two positive real numbers $x,y$, we have $$ \sqrt{xy}≤ \frac{x+y}{2} .$$ Furthermore, equality occurs if and only if $x = y$.  Any and all help would be appreciated.
How to overcome Acrobat Reader error 131 with a pdfLaTeX doc? I'm generating a PDF document with pdflatex (more precisely, latexmk invoked from TexMaker 3.4. My LaTeX install is TeXLive 20120719). I can read the document fine in Okular and Gmail's attachment preview, but a colleague that runs Acrobat Reader on Windows reports: "There was a problem reading this document (131)".  EDIT: The error disappears when using \usepackage[demo]{graphicx}, which puts black boxes instead of the real images. They are all PDFs, PNGs and JPGs. Instead of using demo, I added the code for a \demoincludegraphics command, which makes the file work in Reader when using it to replace the logo in the headers, as shown below. But enabling it (using \includegraphics instead of \demoincludegraphics) and removing everything else also works in Reader! There is some strange global interaction that makes the file appear broken to Reader when enabling all the graphics...  My code compiles fine, reason why I only attach the preamble and a simple text for a Minimal Working Example:  \documentclass[12pt,lettersize,twoside]{article}  \usepackage[utf8]{inputenc} \usepackage[spanish,es-tabla]{babel}   \usepackage[fixlanguage]{babelbib} \usepackage{amssymb,amsmath} \usepackage{graphicx}    \usepackage{array}    \usepackage{caption} \usepackage{subfig} \usepackage{rotating} \usepackage{booktabs} \usepackage{multirow} \usepackage{calc} \usepackage{stfloats} \usepackage{fixltx2e} \usepackage{url} \usepackage[colorlinks=true,linkcolor=blue,citecolor=red]{hyperref} \usepackage[all]{hypcap}  % Margins \usepackage[top=2cm, bottom=2cm, left=3cm, right=2cm]{geometry}  % Make header show up in the first page \makeatletter \renewcommand\maketitle{\par \begingroup     \renewcommand\thefootnote{\@fnsymbol\c@footnote}%     \def\@makefnmark{\rlap{\@textsuperscript{\normalfont\@thefnmark}}}%     \long\def\@makefntext##1{\parindent 1em\noindent             \hb@xt@1.8em{%                 \hss\@textsuperscript{\normalfont\@thefnmark}}##1}%     \if@twocolumn     \ifnum \col@number=\@ne         \@maketitle     \else         \twocolumn[\@maketitle]%     \fi     \else     \newpage     \global\@topnum\z@   % Prevents figures from going at top of page.     \@maketitle     \fi     \thispagestyle{fancy}\@thanks % was {empty} \endgroup \setcounter{footnote}{0}% \global\let\thanks\relax \global\let\maketitle\relax \global\let\@maketitle\relax \global\let\@thanks\@empty \global\let\@author\@empty \global\let\@date\@empty \global\let\@title\@empty \global\let\title\relax \global\let\author\relax \global\let\date\relax \global\let\and\relax } \makeatother  % Remove date field in title; Remove preceding vertical space \makeatletter \renewcommand{\@maketitle}{ \vspace{-3cm} \begin{center}% {\LARGE \@title}\\ \vspace{2mm}% {\@author} \end{center}% \par} \makeatother \makeatletter  % Demo includegraphic boxes \newcommand{\demoincludegraphics}[2][]{%   \setlength{\fboxsep}{-\fboxrule}% Remove frame separator/gap   \framebox{\rule{0pt}{100pt}\rule{150pt}{0pt}}% Framed box of 150pt x 100pt }  % Headers Config  \usepackage{fancyhdr} \pagestyle{fancy} \fancyhead{} \renewcommand{\headrulewidth}{0.4pt} \setlength{\headheight}{2.2cm}  \addtolength{\textheight}{-1.62cm} \fancyhead[LO,RE]{\demoincludegraphics[height=1.6cm]{logo-slogan}} \fancyhead[LE,RO]{}  % Footers Config \fancyfoot{} \renewcommand{\footrulewidth}{0pt} \fancyfoot[C]{\thepage}  % Notas al pie etiquetadas por simbolos   \renewcommand{\thefootnote}{\fnsymbol{footnote}}  \title{Doc Title} \author{My name}  \begin{document}  \maketitle  This is a test. It includes graphics and other floats.   \end{document}   EDIT: I found a workaround: Processing the generated pdf file through Ghostscript, as follows:   gs -dSAFER -dBATCH -dNOPAUSE  -sDEVICE=pdfwrite -sOutputFile=output.pdf input.pdf   where input.pdf is the file generated by pdfLaTeX, and output.pdf the one that will be readable by Adobe Reader and the rest of the PDF readers. Run command on another(new) terminal window How can one express frequentist p-value in terms of Bayesian concepts? Why is there conservation of kinetic energy in elastic collision and not in inelastic collision? Performance differences between equal (=) and IN with one literal value Remove password from an Excel Document I'm providing internal support and one of our users has managed to put a password on an excel file by accident, I've done the proper checks to make sure that the user should have access to the document and now want to know what the recommendation for removing a password from an Excel document.  For what its worth, the password appears after Excel opens but before you can see any data in excel. Illustrator: Is there an efficient and accurate way to join two circles by their tangents (and keep it editable)? It's easy to join two circles if they are the same size and horizontal/vertical, into a pill shape:    If the circles are at an add angle then it's still possible but you can no longer rely on smart guides.     You can add lines roughly where the tangents should be and then use the scissors tool on the circles, cut out the inner area and then join the two crescents into one shape.    Is there a better way to do this? Is there a way to do this that is still editable afterwards too?  There are some other questions that cover the same area as this but don't cover editability or joining circles that are not in line with each other (where smart guides can't help). Examples of patterns that eventually fail Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of number of edges on a complete graph starts $0,1,3,6,10$, so the next term must be 15 etc."  Granted, this second statement is less logically unsound than the first since it's not difficult to see the reason why the sequence must continue as such; nevertheless, the statement was made on a premise that boils down to "interesting patterns must always continue".  I try to counter this logic by creating a ridiculous argument like "the numbers $1,2,3,4,5$ are less than $100$, so surely all numbers are", but this usually fails to be convincing.  So, are there any examples of non-trivial patterns that appear to be true for a large number of small cases, but then fail for some larger case? A good answer to this question should:   be one which could be explained to the layman without having to subject them to a 24 lecture course of background material, and have as a minimal counterexample a case which cannot (feasibly) be checked without the use of a computer.   I believe conditions 1. and 2. make my question specific enough to have in some sense a "right" (or at least a "not wrong") answer; but I'd be happy to clarify if this is not the case. I suppose I'm expecting an answer to come from number theory, but can see that areas like graph theory, combinatorics more generally and set theory could potentially offer suitable answers. I am trying to boot Ubuntu on my computer.  When I boot Ubuntu, it boots to a black screen.  How can I fix this? Table of Contents: Are collective nouns always plural, or are certain ones singular? Fluent and Query Expression — Is there any benefit(s) of one over other? LINQ is one of the greatest improvements to .NET since generics and it saves me tons of time, and lines of code.  However, the fluent syntax seems to come much more natural to me than the query expression syntax.  var title = entries.Where(e =&gt; e.Approved)     .OrderBy(e =&gt; e.Rating).Select(e =&gt; e.Title)     .FirstOrDefault();  var query = (from e in entries              where e.Approved              orderby e.Rating              select e.Title).FirstOrDefault();   Is there any difference between the two or is there any particular benefit of one over other? Has there been any progress made on granting a developer account for SF and HubExchange Devs? It feels conceptually wrong to me a SF product doesn't allow developers to perfect and develop a wider solution offering without paying +10K license fee per year. Even without CRM the HubExchange market could be a richer portal for consumers and developers if we were allowed a means to test our concept.  I think we'd all be perfectly happy with an account that was blackholed or limited in such a way where we could only send a finite qty of emails to an email address associated to the user.  If there's a forum or avenue where we can assist in raising the value proposition please let us know as I know there are many people looking. What's the difference between a temp table and table variable in SQL Server? Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$ I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a &gt;0$$  Which function/ contour should I consider ? WPF DataGrid : CanContentScroll property causing odd behavior What logical fallacy considers a personal experience to be common place While my wife and I were discussing the news a few days ago I stated  that an argument expressed (in a news article, not by my wife) was a fallacy -- but I've been unable to put a name to it.  Basically, it is when a person experiences some experience (personally or thru acquaintance), and thus considers it a common experience because they have experienced it. I think I've got it down to the most basic case:  int main(int argc, char ** argv) {   int * arr;    foo(arr);   printf("car[3]=%d\n",arr[3]);   free (arr);   return 1; }  void foo(int * arr) {   arr = (int*) malloc( sizeof(int)*25 );   arr[3] = 69; }   The output is this:  &gt; ./a.out   car[3]=-1869558540  a.out(4100) malloc: *** error for object 0x8fe01037: Non-aligned pointer                          being freed  *** set a breakpoint in malloc_error_break to debug &gt;   If anyone can shed light on where my understanding is failing, it'd be greatly appreciated. I often see questions edited into this form:     Lorem ipsum dolor sit amet blah blah blah paragraph paragraph original question.            UPDATE: The Lorem disappeared and left this message: "new information".   I find this format unaesthetic (since Stack Exchange posts are supposed to be useful in the long term and update boundaries are artifacts of the creation rather than the content) and feel that new information should, whenever possible, be incorporated into the original text “as if it had always been that way” if practical, and pointing out an update should be done mostly in comments.  Is this an appropriate writing style, and if so under what circumstances? If not, should it be removed or discouraged? One Object won't Show up in Final Render even though it shows up in the Preview Render I have a model of a building with a sidewalk around it.  Both show up when I switch to rendering mode in both Blender Render and Cycles Render mode.  But:  when I hit F12 for a final proper render, the building shows up -- but the sidewalk doesn't.    I've checked the normals and I thought the normals seemed pointed in the right direction.  I've set the Clipping on the Camera to a high number.  I removed all the materials from the sidewalk and added a new material-- the problem persists.  I've exported the blend as an FBX file and the Sidewalk shows up fine in Unity.  Just in final renders, it doesn't show up.    Here is a blend file: .  I tried to destroy everything unrelated to the issue and label things in a clear way.    Thanks. Up my street and down the lane Possible Duplicate:          Except where there is obvious difference in elevation e.g. on a sloping road, how do native speakers decide whether to use up or down in phrases like up the road when referring to a location along the same road?