Let $x,y,z \in \Bbb R^+$ such that $x+y+z=3$. Prove the inequality   $\sqrt x+\sqrt y+\sqrt z \ge xy+yz+zx$     I tried to prove that $\sqrt x+\sqrt y+\sqrt z-(xy+yz+zx)\ge 0$ I squared the equality, put the value of $xy+yz+zx$ (in terms of $x^2+y^2+z^2$). Now I tried to prove that the above expression but failed.    Thanks for hints or solutions.
Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$.  Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$    Here's my idea:  $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(ab + bc + ca)$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) - 2(ab + bc + ca) \ge 0$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) - ((a+b+c)^2 - (a^2 + b^2 + c^2) \ge 0$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) - (a+b+c)^2 \ge 0$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge (a+b+c)^2$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3^2 = 9$  And I'm stuck here.  I need to prove that:   $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge (a+b+c)^2$ or  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3(a+b+c)$, because $a+b+c = 3$  In the first case using Cauchy-Schwarz Inequality I prove that:  $(a^2 + b^2 + c^2)(1+1+1) \ge (a+b+c)^2$  $3(a^2 + b^2 + c^2) \ge (a+b+c)^2$  Now I need to prove that:   $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3(a^2 + b^2 + c^2)$   $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(a^2 + b^2 + c^2)$   $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge a^2 + b^2 + c^2$   I need I don't know how to continue.  In the second case I tried proving:  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(a+b+c)$ and  $a^2 + b^2 + c^2 \ge a+b+c$  Using Cauchy-Schwarz Inequality I proved:  $(a^2 + b^2 + c^2)(1+1+1) \ge (a+b+c)^2$  $(a^2 + b^2 + c^2)(a+b+c) \ge (a+b+c)^2$  $a^2 + b^2 + c^2 \ge a+b+c$  But I can't find a way to prove that $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(a+b+c)$  So please help me with this problem.  P.S   My initial idea, which is proving:  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3^2 = 9$  maybe isn't the right way to prove this inequality.
Options to hide reputation information on Stack Overflow? $a+b+c =0$; $a^2+b^2+c^2=1$. Prove that: $a^2 b^2 c^2 \le \frac{1}{54}$ If a,b,c are real numbers satisfying $a+b+c =0; a^2+b^2+c^2=1$.  Prove that $a^2 b^2 c^2 \le \frac{1}{54}$. "Print" and "ui.Label" return different strings in console and in the UI in Google Earth Engine I am working on a variable label in Google Earth Engine and noticed that the label as printed in the console is not the same as the label shown on the map. Why is that? 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In my Master thesis, I would like to have the title page without any page number and an acknowledgments section and the table of contents with roman numbers. The rest of the document should be numbered arabic.  I have tried this, to have roman numbers for the Table of contents and normal numbers for the rest.  \documentclass[12pt,ngerman,openany]{report} \usepackage{titlesec} \usepackage{titletoc} \usepackage{cite} \usepackage[a4paper]{geometry} %\usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage[ngerman]{babel} %\usepackage{ngerman}{babel} \usepackage[T1]{fontenc} \usepackage{graphicx} \usepackage{float} \selectlanguage{ngerman} \usepackage{amssymb} \usepackage{amsmath} \usepackage{url} \usepackage{epstopdf} \usepackage{scrhack} \usepackage{listings} \usepackage{caption} \titleformat{\chapter}{\rmfamily\huge}{\textbf\thechapter.}{25pt}{\huge\textbf} \defcaptionname{ngerman}{\lstlistingname}{Code} \defcaptionname{ngerman}{\lstlistlistingname }{Codeverzeichnis}  \lstloadlanguages{R}  \lstset{% general command to set parameter(s)    basicstyle=\sffamily\footnotesize, % print whole listing small    keywordstyle=\sffamily\footnotesize\bfseries, % ubold black keywords    identifierstyle=, % nothing happens    commentstyle=\sffamily\footnotesize\slshape, % green comments    stringstyle=\sffamily\footnotesize, % typewriter type for strings    showstringspaces=false, % no special string spaces    numbers=left,    numberstyle=\sffamily\footnotesize,    stepnumber=1,    numbersep=10pt,    showspaces=false,    showtabs=false,    frame=lines,    morecomment=[l]{\%},    float=htbp,    numberbychapter=true  }   \begin{document} \let\cleardoublepage\clearpage   \pagenumbering{roman}  \tableofcontents  \clearpage  \vspace*{75mm} \chapter{Einleitung} \section{Motivation} bla bla bla \section{Ziel}  bla bla \chapter{Stand der Technik zur Mustererkennung} \section{Die Zeitreihen} bla bla \subsection{Beschreibung} bla bla \subsection{Erklärung}  bla bla  \subsection{Vorhersage} bla bla \subsection{Kontrolle} bla bla \section{Feature Extraktion} bla bla \subsection{Autokorrelation} \subsection{Was ist Autokorrelation?} bla bla   \subsection{Autokorrelation und Zeitreihen} bla bla  \subsubsection{Zeitreihe-Diagramm}  bla bla \subsubsection{Lagged Streudiagramm} bla bla  \subsubsection{Autocorrelation-Function (Korrelogramm)} bla bla \subsection{Spektralanalyse} \subsection{Was ist Spektral-Analyse} bla bla \end{document} Who is the Lord, as referred to in the title? 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Proof on showing if F(x,y,z)=0 then product of partial derivatives (evaluated at an assigned coordinate) is -1 The task is as follows:     Given: $$F(x,y,z) = 0$$    Goal:  Show    $\frac{\partial  z}{\partial y}|_x \frac{\partial  y}{\partial x}|_z \frac{\partial x}{\partial z} |_y = -1$   Here is my work so far:  (1) Differentiate with respect to y, I get:  $0 + F_2 + F_3  \frac{\partial z}{\partial y} = 0$  So $ F_3  \frac{\partial z}{\partial y} = - F_2$  (2) Differentiate with respect to x, I get:  $F_1 + F_2 \frac{\partial y}{\partial x}  + 0 = 0$  So $ F_2  \frac{\partial y}{\partial x} = - F_1$  (3) Differentiate with respect to z, I get:  $F_1  \frac{\partial x}{\partial z}    + 0  + F_3 = 0$  So $ F_1  \frac{\partial x}{\partial z} = - F_3$  (4) After some manipulations with the $F_i$, I get to the conclusion that $\frac{\partial z}{\partial y}* \frac{\partial y}{\partial x} * \frac{\partial x}{\partial z}  = -1$, so when evaluated with x, z, y respectively, conclusion is still true  My questions are:  (1) Is my proof correct?  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Error: class X is public should be declared in a file named X.java