What are the maximum sizes of rectangle that can placed into elipse.? please, I have got elipse which is definied $4x^2+9y^2=36$ and I should determine the sizes of $a,b$ so that the rectangle placed into elipse has maximum area. Thanks for help.
Find the area of largest rectangle that can be inscribed in an ellipse The actual problem reads:      Find the area of the largest rectangle that can be inscribed in the ellipse    $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$   I got as far as coming up with the equation for the area to be $A=4xy$ but then when trying to find the derivative I don't think I'm doing it right.
Can I use Divine Intervention or Wish to regain Wish? After reading this question on , I started wondering if there are ways for them to regain that ability.  I figured a Divine Intervention is probably strong enough as that is a god's power, as well as a second caster casting wish to undo the first wish (and thereby undoing its effects).  Are these valid ways of regaining the ability to cast wish? When I have .txt files or .py files, I can Quick Look those files in Finder. I have .jl files that are codes written in the Julia Language. Of course they are text files. They will open in a text editor like Sublime Text. How can I Quick Look these .jl files in Finder? Which one is correct: "wish I was here" or "wish I were here"? Possible Duplicate:          Which one is the correct form: "Wish I was here" or "wish I were here"? I've heard both of them many times but I don't know which one is correct. Infinite metric space has open set $U$ which is infinite and its complement is infinite Let $(X,d)$ be a metric space where $X$ is an infinite set. Prove that the space has an open set $U$ such that both $U$ and its complement are infinite sets.   I have considered if $d$ is the discrete metric, then one such $U$ exists. But if $d$ is not discrete, how do we find such $U$? How to export/backup Notes in macOS in one go I have a lot of Notes on my machine which I don't want to loose. It's not logical to export them one by one. Is there any way to export them in one go, which would make me either use them on my new   I learned that I could backup ~/Library/Group Containers/group.com.apple.notes but I am not really sure if this would be helpful. This is what I get as for this directory.  $ du -sh ./* 4.0K    ./Library 256K    ./NoteStore.sqlite  32K    ./NoteStore.sqlite-shm 3.0M    ./NoteStore.sqlite-wal   I am running macOS Mojave. Are DSLRs a dying breed, making now the time to switch to a mirrorless camera system? was created on Server Fault, answered on Server Fault, I upvoted some of those answers on Server Fault, and then moved by consensus to Super User.  Now that the question is on Super User, I can upvote the same answers I previously upvoted on Server Fault. How is it, that $\sqrt{x^2}$ is not $ x$, but $|x|$? If $a$ and $b$ are relatively prime, prove that $a + 2b$ and $2a+b$ are also relatively prime or have a $\gcd$ of $3$.  I'm very new to number theory so please don't assume I am familiar with some of the terminology. Matlab - Function taking no arguments within a class Rename ToC, List of Figures, References I would like to know how to rename the standard list of figures, table of contents, and references. What I saw so far relying on \renewcommand{\contentsname}{name} did not work so far.  Here's the list of preambles I use:  \documentclass[12pt]{report} \usepackage[utf8]{inputenc}  \usepackage{fourier} \pagestyle{headings} \usepackage{marvosym} \usepackage[pdftex]{graphicx} \usepackage[margin=2.5cm]{geometry} \usepackage{rotating} \usepackage{amsmath} \usepackage{booktabs} \usepackage{url} \usepackage{natbib} \usepackage{setspace} \linespread{1.5} \newenvironment{remark}{\par\mbox{}\par\noindent{\bfseries             \emph{Remark}\/}:\normalfont}{\par} \newenvironment{defin}{\par\mbox{}\par\noindent\bfseries \emph{\Pointinghand\     }\normalfont}{\par} \setcounter{toc**strong text**depth}{2}  \usepackage[colorlinks=true, a4paper=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref} \pdfcompresslevel=9 \setstretch{1.0} %vertical stretch \usepackage{fontspec} \setmainfont[BoldFont       = AvenirNextRoundedPro-Bold.otf , ItalicFont     =AvenirNextRoundedPro-Italic.otf]{AvenirNextRoundedPro-Reg.otf} \defaultfontfeatures{Mapping=tex-text} \usepackage{multirow} \usepackage{array} \usepackage{booktabs} \usepackage{hyperref} \hypersetup{ hidelinks, } \usepackage{ragged2e} \usepackage{euler} \usepackage[euler-digits,euler-hat-accent]{eulervm} \usepackage[labelfont=bf]{caption} \usepackage{wrapfig} \usepackage{amsmath} \usepackage{amssymb}  \usepackage[english]{babel} \usepackage[nottoc]{tocbibind} %Adds "References" to toc \usepackage{titlesec} \titleformat{\chapter}[display]{\bfseries\huge}{\filleft\Large\chaptertitlename~\thechapter}{1ex}{\vspace{1ex}\filright}[\vspace{0.5ex}\titlerule] \usepackage{float} \usepackage{etoolbox} \makeatletter \patchcmd{\@caption}{\csname the#1\endcsname}{\csname fnum@#1\endcsname:}{}{} \renewcommand*\l@figure{\@dottedtocline{1}{2em}{6em}} \let\l@table\l@figure  \makeatother % default for 3rd arg: 2.3em - for list of figures and tables \usepackage{background} \definecolor{textcolor}{HTML}{000000}             \newcommand\Text{c\quad o\quad n\quad f\quad i\quad d\quad e\quad n\quad t\quad i\quad a\quad l} \SetBgColor{textcolor}\SetBgOpacity{0.3}\SetBgAngle{90}         \SetBgPosition{current page.center} \SetBgVshift{-0.33\textwidth} \SetBgScale{1.8}         \SetBgContents{\sffamily\Text} %for confidential   I also use TexMaker. I do not know how relevant it is, but here my document language is set to English (even if it's not the language I write in, in that document, but it isn't a problem seemingly). What I also saw is that, if I remove babel and the references, I can rename my ToC and LoF, LoT. But I need the references though. Any tip would be very welcome!  Edit: I cannot provide text data from my project, because it is confidential. Here's what I can provide from the main document. Here is what I can include more.  \begin{document} \include{02_front}  \setcounter{page}{0} \pagenumbering{roman} \tableofcontents \listoffigures \listoftables \newpage \setcounter{page}{0} \pagenumbering{arabic} \chapter{blabla} \section{test é è åøæ î} \appendix \bibliographystyle{plain}  \bibliography{80_refs}   \end{document}   Also, I actually finished writing my thesis but I need to summarize it briefly in another language, am would be extremely unwilling to test and remove packages that "might be" not very useful. I literally only have to change these names of ToC, LoF, ToF and references. Searching through ODT documents without opening them? Let's say I have 2000 .ODT files. Their names are random numbers. How do I go about searching for, let's say, "pricing list"? How do I find the document I need without opening all of them one by one and checking to see if it's the right one?  Is there some program to search through the words in the documents without opening them? Today is the release of Ubuntu 21.04 (Hirsute Hippo). And I can't update to this version. And also I selected ‘Notify me of a new Ubuntu version’ and changed it to ‘For any new version' on the Updates tab on Software &amp; Update, but it's still not working. In addition, in the terminal, I've written: sudo do-release-upgrade -c  And output was: Checking for a new Ubuntu release No new release found.  What should I do? When is a biased estimator preferable to unbiased one? I just fit a model in lme4, and I'm wondering what the heck I fit...  I have individuals id, and each is measured pass/fail on items that can be described using two factors, f1 and f2. My theory says that id and f2 can interact. So I want to compare these models:  # f2 doesn't matter m1 &lt;- lmer(pass ~ f1 + (1|id), family="binomial")  # f2 is a fixed effect, doesn't interact with individuals m2 &lt;- lmer(pass ~ f1 + f2 + (1|id), family="binomial")  # f2 is a nested random effect, interacting with individuals m3 &lt;- lmer(pass ~ f1 + (1|f2/id), family="binomial")   First, am I right that there's a random effect of each level of f2, and each individual's level of f2 is partially pooled, depending on the number of observations for that individual?  Second, does it make sense to fit f2 as a fixed effect with random individuals nested within each level of f2? How does one write that?  Third, what's this, with f2 and id reversed in the formula? I typed it by accident, but now I want to understand it.  m3weird &lt;- lmer(pass ~ f1 + (1|id/f2), family="binomial") Can I use Stack Exchange logos on a t-shirt or other apparel? I happened to notice while on the Super User site that one user's picture shows him wearing a shirt with the Super User logo on it. Interested, I wanted to find out where he got it from. Looking through old questions on Meta, I found out that there used to be a shop.stackexchange.com site that sold Stack Exchange gear. Unfortunately, they no longer sell any merchandise.  Is there any site where I can purchase a Stack Exchange site shirt/hat/etc.? If not, could I create a shirt/hat/etc. myself with one of the sites' logos on it? If so, where can I locate the logos in their largest size? How to get the current user in ASP.NET MVC Here's an example of a Google Docs Spreadsheet script that sends emails using Google spreadsheets. I want the email Sent in HTML not plain text. Can someone help me out?  function sendEmails() {   var sheet = SpreadsheetApp.getActiveSheet();   var startRow = 2;  // First row of data to process   var numRows = 2;   // Number of rows to process   // Fetch the range of cells A2:B3   var dataRange = sheet.getRange(startRow, 1, numRows, 2)   // Fetch values for each row in the Range.   var data = dataRange.getValues();   for (i in data) {     var row = data[i];     var emailAddress = row[0];  // First column     var message = row[1];       // Second column     var subject = "Sending emails from a Spreadsheet";     MailApp.sendEmail(emailAddress, subject, message);   } } Why do video cables (DVI, VGA) still screw in? I guess I could imagine that there was a time when accidentally disconnecting the monitor would have been catastrophic, though I don't see that being the case today.  Is it just a legacy holdover that they are still secured with screws?  Also, why is it that the video connection is really the only one that is still secured this way?  It seems like the power cable would be a better candidate to be secured with screws.  That really could be catastrophic.  In the end, the reason this question occurred to me is that I've never had a video cable come out when not screwed in, so why the significant security? Prove $\sum_{i=1}^n i! \cdot i = (n+1)! - 1$? Prove the summation:  $$\sum_{i=1}^n i! \cdot i = (n+1)! - 1$$ using induction.   base case:  $n=1$: \begin{align*} \sum_{i=1}^1 i! \cdot i &amp;= (1+1)! - 1 \\ 1 &amp;= 2 - 1 \\ 1 &amp;= 1 \end{align*}   This is a question from my test review packet, currently have the base case completed and I am a bit lost on where to go from there. Any help/hints are appreciated.