How can we show that the equality doesn't occur? Let $f:[0,1] \to [1,2]$, continuous on $[0,1]$ with $1&lt;f(1)&lt;2$. Show that $$\int_0^1\left (f^2(x)-2f(x)\right )\, dx&gt;-1.$$ We have that $$(f(x)-1)^2\geq 0\implies \int_0^1 (f(x)-1)^2\, dx\geq 0 \Rightarrow \int_0^1\left (f^2(x)-2f(x)\right )\, dx\geq -1.$$ Till here we have $\geq$ instead of $&gt;$. How can we show that the equality doesn't occur? Does the same holds also if $f$ is Riemann integrable?
Show that if a function is not negative and its integral is $0$ than the function is $0$ Suppose $f$ is continuous, $f(x) \ge 0$ on $[a,b]$ and $\int_a^b f(x) dx=0$. Show that $f(x)=0$ on $[a,b]$.   I know that if $f$ in continuous at $x_0$ and $f(x_0) &gt; 0$, then there exists $\delta &gt; 0$ s.t. $f(x) \ge \frac{1}{2} f(x_0)$ for $|x-x_0| &lt; \delta$ and I think it should help me but I'm stuck at this point.
Installing Wine on 64 Bit I have tried to install wine through the software center and terminal and I get this error.  chris@ubuntu:~$ sudo apt-get install wine1.5 [sudo] password for chris:  Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  wine1.5 : Depends: wine1.5-i386 (= 1.5.16-0ubuntu1) but it is not installable            Recommends: gnome-exe-thumbnailer but it is not going to be installed or                        kde-runtime but it is not going to be installed            Recommends: ttf-droid            Recommends: ttf-mscorefonts-installer but it is not going to be installed            Recommends: ttf-umefont but it is not going to be installed            Recommends: ttf-unfonts-core but it is not going to be installed            Recommends: winbind but it is not going to be installed            Recommends: winetricks but it is not going to be installed E: Unable to correct problems, you have held broken packages. Matrix algebra: If $A^2=0$, Proof rank(A) $\le \frac{n}{2}$ Filter data.frame rows by a logical condition Let $\varphi:G\rightarrow H$ be a group homomorphism from group $G$ to group $H$. Show that, if $H$ is abelian, all subgroups of $G$ that contain $\mathrm{ker} (\varphi)$ are normal in $G$. I've rooted my phone.  Now what?  What do I gain from rooting? This is a common question by those who have just rooted their phones.  What apps, ROMs, benefits, etc. do I get from rooting?  What should I be doing now? The title says it all.  If I was on a bus at 60 km/h, and I started walking on the bus at a steady pace of 5 km/h, then I'd technically be moving at 65 km/h, right?  So my son posed me an interesting question today: since light travels as fast as anything can go, what if I shined light when moving in a car?  How should I answer his question? How do you read C declarations? I have heard of some methods, but none of them have stuck. Personally I try to avoid complex types in C and try to break them into component typedef.  I'm now faced with maintaining some legacy code from a so called 'three star programmer', and I'm having a hard time reading some of the ***code[][].  How do you read complex C declarations? Giving a single user write permissions I want to grant a single user write permissions to a directory owned by root over command line. With chmod, it seems that I can only change the permissions for owner, group and other, not for a specific user. Show that $d_2$ defined by $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric How iptables tables and chains are traversed I know linux has 3 built-in tables and each of them has its own chains as follow:  FILTER: PREROUTING, FORWARD, POSTROUTING  NAT: PREROUTING, INPUT, OUTPUT, POSTROUTING  MANGLE: PREROUTING, INPUT, FORWARD, OUTPUT, POSTROUTING  But I can't understand how they are traversed, in which order, if there is. For example, how are they traversed when:   I send a packet to a pc in my same local network when I send a packet to a pc in a different network when a gateway receives a packet and it has to forward it when I receive a packet destinated to me any other case (if any) Why is my document pdf generated by pdflatex looks different from the one generated by latex+dvips+ps2pdf ? From the code below, i have tow different papers switch to the tow cited commands above :  \documentclass[10pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[francais]{babel} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{graphicx} \usepackage{lipsum} \author{M ROUAN SERIK} \title{Article with \LaTeXe} \fontfamily{cmr} \begin{document} \maketitle \section{Titled section} \lipsum \end{document}   Here's my pdffonts for the tow pdfs : Pdflatex:    name                   type     encoding  emb sub uni object ID   ------------------------------- ---------------- --- --- --- ---------   AMTRHA+SFRM1728        Type 1    Custom   yes yes no       4  0   DPTCLU+SFRM1200        Type 1    Custom   yes yes no       5  0   JVZRKA+CMMI12          Type 1    Builtin  yes yes no       6  0   UEWNID+SFBX1440        Type 1    Custom   yes yes no       7  0   FOVERL+SFRM1000        Type 1    Custom   yes yes no       8  0     latex+dvips+ps2pdf      name                                 type              encoding         emb sub uni object ID   ------------------------------------ ----------------- ---------------- --- --- --- ---------   WVMUEH+SFRM1000                      Type 1C           WinAnsi          yes yes no        16  0   ZHRBNN+SFBX1440                      Type 1C           WinAnsi          yes yes no      14  0   IMJWCD+CMMI12                        Type 1C           Custom           yes yes no      12  0   MDMSVR+SFRM1200                      Type 1C           WinAnsi          yes yes no      10  0   TWHNMT+SFRM1728                      Type 1C           WinAnsi          yes yes no       8  0 Is there an idiom available, that is exactly opposite to "Cake walk" or "Child's play"? What is terminal velocity? I've heard the term especially when the Discovery Channel is covering something about sky diving. Also, it is commonly known that HALO (Hi-Altitude, Lo-Opening) infantry reaches terminal velocity before their chutes open.    Can the terminal velocity be different for one individual weighing 180 pounds versus an individual weighing 250 pounds? How do I tell if ACR will work with my camera and how do I get the correct version for Photoshop? "The queen of England's crown" or "The queen's of England crown"? Possible Duplicate:          What is the correct way of these two sentences?   The queen of England's crown The queen's of England crown   Strictly linguistically, sentence 2 should be the correct one, since the crown belongs to the queen and not to England but it sounds really awkward to pronounce. Record a macro of windows commands Using Windows 7, is there an in-built or third-party method of recording windows commands and being able to play them back?  As a very basic example of what I would like to be able to do, imagine having two desktop themes that you like to alternate between. The Windows commands would be something like   Open personalisation applet select second theme ok   (I'm calling this a macro because of the equivalent capability in Word.) Gradient and Swiftest Ascent I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$)  The standard proof is to note that the directional derivative is $$D_vf=v\cdot \nabla f=|\nabla f|\,\cos\theta$$ which is maximized at $\theta=0$. This is a good verification, but it doesn't really help me understand the result. Space of bounded continuous functions is complete As you can see in object mode there are 4 vertices that have separated the front of the cartridge into multiple faces when I use boolean modifiers on cartridge.001. How can I fix this?  I imagine it has to do with good topology.  Here's the cartridge in object mode showing faces:    Here is the cartridge with the unwanted vertices:    Attached is the .blend Error when installing: "unable to find a medium containing a live file system" Error message:  (initramfs) unable to find a live medium containing a live file system   I got this error while trying to install Ubuntu 10.10 from a bootable USB stick on to Asus X64J series laptop. After getting this error the installation fails to start.  I have used the same USB stick on some other laptops and the installation started as usual. Any help will be appreciated.