Let $f:[a,b] \mapsto \mathbf{R}$ be a continuous function such that $f(x) \geq 0$ for all x $\in [a, b].$ Suppose $\int_a^b f = 0.$ Show that $f(x) = 0$ for all x $\in [a,b]$
Let $a &lt; b$ and $f:[a, b] \rightarrow \mathbb R$ be a Riemann integral function. Suppose that $f(x) \geq 0$ for all $x \in [a, b]$. I want to show whether $\int_a^bf(x) = 0$ implies $f(x) = 0$ or not, but I'm not sure where to start. I don't know much about Riemann integral functions in general. I only know that $$\underline \int_a^b f= \overline \int_a^b f.$$ Also, I know the fact that if $f$ is continuous, then $f(x) = 0$ for all $x\in [a, b]$. Can anyone provide any hints for me, because I'm really confused about this whole thing.
According to two questions  and  asked on this site earlier there exists a nice relation:  $$\frac1{n!} \left(\int_{0}^t\mathrm dt \; f(t)\right)^n = \int_{0}^t\mathrm dt_1 \int_{0}^{t_1}\mathrm dt_2 \cdots \int_{0}^{t_{n-1}}\mathrm dt_n\; f(t_1)\,f(t_2) \ldots f(t_n). $$  Unfrotunatly I am more interested in integrals of the form  $$\int_{0}^t\mathrm dt_1 \int_{0}^{t_1}\mathrm dt_2 \cdots \int_{0}^{t_{n-1}}\mathrm dt_n\; f(t_n). $$  Can it be somehow brought to an unary integral?  All I have figured out by now is that  $$\int_{0}^t\mathrm dt_1 \int_{0}^{t_1}\mathrm dt_2 \cdots \int_{0}^{t_{n-1}}\mathrm dt_n\; t_n^m = \frac{m!}{(m+n)!} t^{m+n} .$$ Does Ubuntu come with a preinstalled or automatic firewall? If not, do I need one?   I've read some articles about the advantages of Linux over other operating systems about security (no need to have antivirus, ...) but would like to be sure. Question is about Ubuntu release process. Are migrations from n to n+1 releases expected?  Assuming we have server with installed Ubuntu 14.04 LTS. Will it be smoothly upgradable to Ubuntu 16.04 LTS via apt-get upgrade ?  Assumptions:   scope is software therefore we assume that given hardware is supported on both releases (upgrading from and to) that's why, always before upgrade you are asked to make backup to be able to revert in case of failure How do I print all lines of a file with duplicate values in a certain column Given a file like this:  1,768,12,46576457,7898 1,123,435,134,146 2,345,6756856,12312,1311 5,234,567465,12341,1341 1,3245,4356345,2442,13 9,423,2342,121,463 9,989,342,121,1212   I would like to list all rows (in bash terminal) such that the value in column 1 appears at least twice (in column 1). The result should be  1,768,12,46576457,7898 1,123,435,134,146 1,3245,4356345,2442,13 9,423,2342,121,463 9,989,342,121,1212 Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$ Let $l \in\mathbb{R} $ and $ f:(0,\infty) \to \Bbb R $ be a function such that $ \lim_{x\to \infty} xf(x)=l $ .  Prove that  $ \lim_{x\to \infty} f(x)=0 $.  Any help would be appreciated - I found this difficult to prove Let $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $5\}$. Let $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $5\}$. Show that $I$ is an ideal of $\Bbb Z[i]$. Is $I$ a maximal ideal? And to find the numbers of elements of the quotient ring $\Bbb Z[i]/ I$.  I have shown that $I$ is an ideal of $\Bbb Z[i]$. But how to verify that $I$ a maximal ideal or not?   And to find the numbers of elements of the quotient ring $\Bbb Z[i]/ I$.   Is there any general way to verify when  $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $n\}$ will be maximal or not? Copy current terminal prompt to clipboard Is there a way to copy the currently written content of the bash prompt? Say I wrote:  ping www.google.com   so that the lines in terminal looks like:  petr@sova:~$ ping www.google.com   and didn't hit return yet. I want to copy that text to clipboard. Is that possible?  The usual usecase would be hitting up several times and finding a command which I want to copy. What is the maximum number of Locations Discovered that one can achieve in Skyrim? Geometric mean never exceeds arithmetic mean This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.)     The geometric mean of $n$ positive numbers $a_1, a_2,\ldots,a_n$ is $\sqrt[n]{a_1a_2 \ldots a_n}$ and their arithmetic mean is $\frac{a_1+a_2+\ldots+a_n}{n}$. If $a_1, a_2,\ldots,a_n$ are $n$ positive real numbers, prove by induction that their geometric mean is always smaller than or equal to their arithmetic mean, i.e. $\sqrt[n]{a_1a_2\ldots a_n} \leq \frac{a_1+a_2+\ldots+a_n}{n}$ Why are the even and odd Regge trajectories degenerate? React on global hotkey in a Java program on Windows/Linux/Mac? A Java6 application sits in the system tray. It needs to be activated using a hotkey (e.g. Super-G or Ctrl-Shift-L etc) and do something (e.g. showing an input box).  How do I do that on:   Windows (XP or Vista) OS/X Linux (Gnome or KDE) Equivalent Resistance? How do you find the equivalent resistance here? (I realized it's a Wheatstone bridge, but it's unbalanced, so that doesn't help.) I tried using Kirchhoff's laws, but there is no cell given. This is about the seemingly controversial shield to ground connection. The system is basically a body-worn intel-based computer, powered by a battery. Two USB 3.0 (or 3.1 Gen1) cables come out to a PCB that contains two USB 3.0 hub chips (). Two external cameras plug in to this usb hub PCB (one camera for each hub). So the usb hub PCB has 4 usb connectors (2 upstream and 2 downstream).  The question is, what to do with each of the usb connector shields? The main directive is USB connection robustness.  I have seen many recommendations. For example:  Recommendation 1  TI's TUSB8020B hub reference design  ties all shells directly to ground.    Intel's  also recommends to tie to ground (although this was written for USB 2.0).    Recommendation 2  TI's TUSB8020B EVM (and datasheet) connect shields together, and to ground using RC filter:    Microchip's  also ties shields together and uses an RC filter, but a 3 orders of magnitude smaller R:    Recommendation 3  Cypess  ties each shield independently to ground using either LC or L filter (chokes really):    The cameras themselves (off the shelf) use Cypress recommendation (LC to ground). The embedded computer seems to tie to ground from visual inspection and checking continuity to ground, but I'm not 100% sure (schematics are not available).  Now we face the shield dilemma with the hub PCB (which by the way currently does not have a metallic enclosure, it is a 3d printed plastic enclosure).   What is your say? When do I need to invoke \phantomsection? How do I compare strings in Java? I've been using the == operator in my program to compare all my strings so far. However, I ran into a bug, changed one of them into .equals() instead, and it fixed the bug.  Is == bad? When should it and should it not be used? What's the difference? So I am in an introductory linear algebra(with proofs) class, and I came across the following theorem:  If $S$ is a subspace of a vector space $V(F)$ and is generated by $\{a_1,...,a_n\}$ then any subset $\{b_1,...,b_m\}$ of $S$ with $m&gt;n$ is linearly dependent.   The proof in the textbook is done through induction (as most proofs regarding finite sets are) but I really do not follow it and I feel there is a simpler way of doing it. A hint would be appreciated. What's the origin of "I'm down with it"? I understand it's an expression of agreement. What exactly does it mean and where did it originate from? PHP: Right way to declare variable before use in loop I have a variable that is built in loop. Something like:  $str = ""; for($i = 0; $i &lt; 10; $i++) $str .= "something";   If $str = "" is ommitted, I get undefined variable notice, but I thought php auto-declare a variable the first time it sees undeclared one?  How do I do this right? How do I increase my code coverage, or why can't I cover these lines? This is a canonical question and answer developed by the community to help address common questions. If you've been directed here, or your question has been closed as a duplicate, please look through the resources here and use them to shape more specific questions. To browse all canonical questions and answers, including more unit test resources, navigate to the  tag.  This canonical question is intended to address several classes of common questions by providing a quick summary and links to comprehensive resources:   How is code coverage obtained and calculated? How do I increase my coverage?  Why can't I cover these lines? Why isn't the body of my loop or conditional statement covered? Where can I get an Ubuntu Installation CD/DVD/USB? I need to upgrade my existing Ubuntu system, and I also want to install Ubuntu on a new laptop.  I would like to have an installation CD or DVD to do both of these.  Where exactly I can order or request an Ubuntu installation CD/DVD/USB?