Given a function f which is smooth Given a function $f:\mathbb{R}\rightarrow \mathbb{R}$ which is smooth and $f(0)=0,f(1)=1$.  Proof that $\exists  x_1,x_2 \in [0;1] \mspace{7mu} (x_1\neq x_2 )$ such that  $\frac{1}{f'(x_1)}+\frac{1}{f'(x_2)}=2$ .  By the Lagrange Mean value theorem $\exists x_0 \in [0;1]$ such tat $f'(x_0)=1$. If $x_1=x_2=x_0$, it's OK, but $x_1 \neq x_2$.
For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $ Suppose that $f$ is continous on $[0,1]$ , differentiable on $(0,1)$ , and $f(0)=0$ and  $f(1)=1$.For every integer $n$ show  that there must exist $n$ distinct points $\alpha_1,\alpha_2\cdots,\alpha_n$ in that interval so that    $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $   Is it theorem or i can show it with basic differentiation knowledge .Please give me hint .
Outlook: scheduling recurring out-of-office? In Outlook, is there a way (or an add-on that can do it) to schedule a recurring out-of-office message?  I work 4 10's, so I'm out of the office every Monday, and it's a bit of a pain to have to reset my out-of-office message every single week. Let $K=F(x)$ be the rational function field over a field $F$ of characteristic $0$, let $L_1=F(x^2)$, and $L_2=F(x^2+x)$. How to show that $L_1\cap L_2 = F$? It was soft and covered in flies. My guess is fungi.  Location: South Florida. Windows 10 ver 1607: File Explorer long paths not working? I updated to the Windows 10 home anniversary edition version 1607 and set LongPathsEnabled to 1 with regedit.exe but the File Explorer doesn't seem to work with long paths still. If you select a long path file and try to rename it, no option will be available. I restarted the computer after making the changes and update.   Is there any way to enable renaming in the File Explorer in the new updated version? Did I miss a step somewhere? Enable External Sharing Programmatically We have a scenario on which we are creating site collections from a provider-hosted app. Some of these site collections will be used by external users. Is there a way from CSOM (server-side) to enable external sharing on the site collection? I haven't found any resources stating whether this is or not doable.  I'm trying to replicate the settings you can specify at the site collections page on the tenant admin site:    Thanks! How to find stopping distance of a car? How do people who do not speak Indian English "prepone" things? Why does evolution not make life longer for humans or any other species?  Wouldn't evolution favour a long life? How to duplicate a div in JavaScript Why don't virtual particles violate conservation of mass/energy? How to install packages in ubuntu without internet connection? Does a controlled mount share its rider's turn? From "Controlling a Mount" on p.198 of the Player's Handbook:     The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options; Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it.   Does this mean that the rider and mount share a single turn? Intuition about Herbrand Models. I'm having trouble understanding how Herbrand models work. I feel like there is a bigger picture to it which I'm not understanding. Why does every satisfiable first order formula have a herbrand model at all ? Can I somehow construct a Herbrand model from a normal model ? My book also says that : a formula is unsatisfiable iff the Herbrand expansion is unsatisfiable. The book I'm using &quot;Mathematical Logic for computer science by Mordechai Ben-Ari&quot; has all the proofs of the above but its not very intuitive to me. Could somebody share their point of view? Proof of Equivalence Relation $R=\{(a,b)\in\Bbb N^2: 5\mid (2a+3b)\}$ Given a set of natural numbers ($\Bbb N$), and the relation $R=\{(a,b)\in\Bbb N^2: 5\mid (2a+3b)\}$. Is $R$ an equivalent relation?   My answer: Yes, because $2a + 3b$ is divisible by $5$, if and only if, $2a + 3b − 5b$ is divisible by $5$.  Is my reasoning right?   If it is not, then how do I prove that this is an equivalent relation? What is the amplitude of a light wave? Referring to this question   I'm curious about what is a amplitude of a light wave. Especially for light from a thermic source. Performance of Non Clustered Indexes on Heaps vs Clustered Indexes Can't install Oracle JDK 9 I uninstalled jdk packages and now can't install them back. I managed to install oracle-java7-installer and oracle-java8-installer but can't install oracle-java9-installer. It seems that my computer tries to download wrong version of oracle-java9-installer 177 when version i would be able to download is 178.    $ wget http://download.java.net/java/jdk9/archive/178/binaries/jdk-9+178_linux-x64_bin.tar.gz --2017-07-21 11:00:02--  http://download.java.net/java/jdk9/archive/178/binaries/jdk-9+178_linux-x64_bin.tar.gz Connecting to 10.144.1.10:8080... connected. Proxy request sent, awaiting response... 200 OK Length: 293348142 (280M) [application/x-gzip] Saving to: 'jdk-9+178_linux-x64_bin.tar.gz'  jdk-9+178_linux-x64_bin.tar.gz     1%[&gt;                                                          ]   5.42M  5.04MB/s               ^C  $ wget http://download.java.net/java/jdk9/archive/177/binaries/jdk-9+177_linux-x64_bin.tar.gz --2017-07-21 11:00:16--  http://download.java.net/java/jdk9/archive/177/binaries/jdk-9+177_linux-x64_bin.tar.gz Connecting to 10.144.1.10:8080... connected. Proxy request sent, awaiting response... 404 Not Found 2017-07-21 11:00:17 ERROR 404: Not Found.   Can you help? apt-get tries to install Oracle JDK 9.  I run Ubuntu 16.04 in virtual machine hosted by win7  Full error message:  sudo apt-get install oracle-java9-installer Reading package lists... Done Building dependency tree        Reading state information... Done oracle-java9-installer is already the newest version (9b177-1~webupd8~0). 0 upgraded, 0 newly installed, 0 to remove and 147 not upgraded. 1 not fully installed or removed. After this operation, 0 B of additional disk space will be used. Do you want to continue? [Y/n] Y Setting up oracle-java9-installer (9b177-1~webupd8~0) ... Using wget settings from /var/cache/oracle-jdk9-installer/wgetrc Downloading Oracle Java 9... --2017-07-21 10:54:11--  http://download.java.net/java/jdk9/archive/177/binaries/jdk-9+177_linux-x64_bin.tar.gz Connecting to 10.144.1.10:8080... connected. Proxy request sent, awaiting response... 404 Not Found 2017-07-21 10:54:13 ERROR 404: Not Found.  download failed Oracle JDK 9 is NOT installed. dpkg: error processing package oracle-java9-installer (--configure):  subprocess installed post-installation script returned error exit status 1 Errors were encountered while processing:  oracle-java9-installerE: Sub-process /usr/bin/dpkg returned an error code (1) Continuity and existence of directional derivatives at all directions of a $f:\mathbb{R}^{2} \to \mathbb{R}$ given by $f(x,y)=\frac{2x^2y}{x^2+y^2}$ Let $f(x,y)=     \begin{cases}       \frac{2x^2y}{x^2+y^2} &amp; if &amp; (x,y) \neq (0,0) \\       0       &amp; if &amp; (x,y)=(0,0) .  \end{cases}$  (i) Prove the directional derivatives of $f$ exist in any direction at the point $(0,0)$.  (ii)¿Is $f$ continuous on (0,0)?  For (i) I took $u=(u_{1},u_{2}) \in \mathbb{R^{2}}$ such $\|u\|=1$ and $0=(0,0)$. So $$\lim_{t \to 0} \frac{f(0+ t u)-f(0)}{t}=\lim_{t \to 0}\frac{2(tu_{1})^2 (tu_{2})}{(tu_{1})^4+(tu_{2})^2}=\lim_{t \to 0}\frac{(t^2 u_{1}^2)(t^2 u_{1}^{2}) tu_{2}}{t^{4}u_{1}^{4}+t^{2} u_{2}^{2}}.$$ But I cannot find the limit when $t$ aproaches to $0$. Which by finding them I would prove the directional derivative exist at every direction right?  For (ii) I got the intuition $f$ is not continuous at $(0,0)$ so I took $\lbrace (\frac{1}{k},\frac{1}{k}) \rbrace_{k \in \mathbb{N}}$ and $\lbrace (\frac{1}{k},0) \rbrace_{k \in \mathbb{N}}$ which are two different sequences in $\mathbb{R}^{2}$ converging to $(0,0)$. However, $\lbrace f(\frac{1}{k},\frac{1}{k}) \rbrace_{k \in \mathbb{N}}=\lbrace \frac{1}{k^{2}} \rbrace_{n \in \mathbb{N}} \to 0$ and $\lbrace f(\frac{1}{k},0) \rbrace_{k \in \mathbb{N}} \to 0$. So maybe my intuition was not right? Can anyone help me end the proof of continuity or not continuity , please? Distinct digits in a combination of 6 digits SQL-Like Inner Join on IDs in Two Layers in QGIS Using QGIS v2.6.0-Brighton.  This question is not about editing layers geographically in anyway. It is merely trying to find data from two layers as you would in an SQL Query and append an extra ID to the second layer.  I have one polyline layer each identified with two different IDs. There is a point layer also with one of these IDs, I wish to find this corresponding ID and append them as a column to the point layer.  How do I do this without finding each row individually and typing it in manually (i.e. larger scale)? I would prefer to use SQL like syntax but if there is a GUI as well that would be ideal. Happy to install official plugins.