Consider differential function $f:\mathbb{R} \to \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$   we have  $$f(b)-f(a)=(b-a)f'(\frac{a+b}{2})$$  Then show that every such function is polynomial of degree less than or equal to two.  i was thinking of applying Roll theorem. but got nowhere.please  provide a hint   Thanks in advanced
Every differentiable function $f:R \rightarrow R$ with the property that $(2h)f′(x) =f(x+h)−f(x−h)$ for all $x \in R$ and all $h$ has the form $f(x) =ax^2+bx+c$.  I would like to get some help on this one. Taking partial derivatives was an idea, but is that legal?
Warning: Total photoshop newb!  I recently got a cool image of my fiancee and I holding hands and showing off her engagment ring. The backdrop is the beach:    I tried to trace an image using a combo of the laso and magic wand tools to select my target. Then I used that to pull out the target as another layer.  I then pasted in a background gradient.     Ended up with a really lame photoshoped looking picture. Notice the white lines where I didn't get the arms close enough and the choppy spots where I got too close.    So what's my main problem here?   Is their a way to bet a better cutout? Is their a blending tool that will make the background look more natural? both   I tried a dozen or so tutorials I found in google searches but they just don't blend these types of images together in any of them. Thanks in advance. How can I list the contents of a .zip folder in c#? I'm trying, but something is wrong with my code in LaTeX: \begin{asy} pair dynkinMark(real x, real y, string s = &quot;&quot;, pen p = defaultpen, pair B = dir(45)) { if (s != &quot;&quot;) { dot(&quot;$&quot;+s+&quot;$&quot;, (x,y), B, p); }     else { dot((x,y), p); }     return (x,y); }  dynkinMark(0,1, &quot;1&quot;, dir(135)); dynkinMark(0,2, &quot;2&quot;, dir(135)); dynkinMark(0,3, &quot;3&quot;, dir(135)); dynkinMark(0,4, &quot;4&quot;, dir(135)); dynkinMark(7,1, &quot;1'&quot;); dynkinMark(7,2, &quot;2'&quot;); dynkinMark(7,3, &quot;3'&quot;); dynkinMark(7,4, &quot;4'&quot;); path p1 = (0,1)..(2,2.0)..(5,3.5)..(7,4); path p2 = (0,2)..(2,1.2)..(5,2.5)..(7,3); path p3 = (0,3)..(2,2.8)..(5,1.5)..(7,1); path p4 = (0,4)..(2,3.8)..(5,3.5)..(7,2); draw(p1, blue); draw(p2, heavycyan); draw(p3, mediumcyan); draw(p4, lightblue); dot(&quot;$12$&quot;, IP(p1, p2), dir(90), heavygreen); dot(&quot;$13$&quot;, IP(p1, p3), dir(90), heavygreen); dot(&quot;$23$&quot;, IP(p2, p3), dir(90), heavygreen); dot(&quot;$14$&quot;, IP(p1, p4), dir(90), heavygreen); dot(&quot;$24$&quot;, IP(p2, p4), dir(90), heavygreen); real y = 0.8; label(&quot;$s_1$&quot;, (IP(p1,p2).x, y), heavygreen); label(&quot;$s_2$&quot;, (IP(p1,p3).x, y), heavygreen); label(&quot;$s_1$&quot;, (IP(p2,p3).x, y), heavygreen); label(&quot;$s_3$&quot;, (IP(p1,p4).x, y), heavygreen); label(&quot;$s_2$&quot;, (IP(p2,p4).x, y), heavygreen); \end{asy} I want to construct a complex power series with the radius of convergence $R=1$ that diverges in:  1) $k$ points on the circle $\{|z|=1\}$.  2) countable number of points on the circle  I have already crafted series that converges on the whole circle and diverges on the whole circle.  Can anybody help with the problem?  Update: I have googled a series $\sum{\frac{z^{kn}}{kn}}$ that is an answer for 1). Though I still need help for the countable case. Suppose $X, Y$ are independent random variables with $X\sim NB(r,p)$ and $Y\sim NB(s,p)$. Then $$X + Y \sim NB(r+s,p)$$  How do I go about proving this? I'm not sure where to begin, I'd be glad for any hint. The result of ls * , ls ** and ls *** I know using the command ls will list all the directories. But what does the ls * command do ? I used it and it just lists the directories. Does the star in front of ls mean how deep it will list the directories? Can I use multiple regression when I have mixed categorical and continuous predictors? It looks like you can use coding for one categorical variable, but I have two categorical and one continuous predictor variable. Can i use multiple regression for this in SPSS and if so how? thanks! Uncountable ordinals without power set axiom Assume $M$ is a set, in which all axioms of $ZF - P + (V=L)$ hold. Does then $M$ believe that there exists an uncountable ordinal? I mean, why should the class of all countable ordinal numbers be a set? How can I put the computer to sleep from Command Prompt/Run menu? Accessing Email logs using REST or SOQL How do I write a correct micro-benchmark in Java? Creating a node fitting the horizontal width of two other nodes I have the two PGF nodes foo and bar positioned in a row.  \documentclass{article} \usepackage{tikz} \usetikzlibrary{positioning} \begin{document}   \begin{tikzpicture}[node distance=0.2cm,mynode/.style={rectangle,draw}]     \node[mynode] (foo)                {foo};     \node[mynode] (bar) [right=of foo] {bar};   \end{tikzpicture} \end{document}   How can I create a third node below those two nodes with the width of foo, bar and the node distance between them? Is there a way to do this with the fit library or is this the wrong approach? Trigonometry question finding the sin and tan If angle $\alpha$ is reflex, and $\cos \alpha = -\frac{9}{41}$, without using a calculator, evaluate  (a) $\sin \alpha$ (b) $\tan \alpha$ (c) $\cos (\alpha - 180\deg)$     . Equal and opposite reaction - forces coming in pairs - or just common sense? Can anyone explain IEnumerable and IEnumerator to me? Can anyone explain IEnumerable and IEnumerator to me? For example, when to use it over foreach? what's the difference between IEnumerable and IEnumerator? Why do we need to use it? What does “{} \;” mean in find (in Linux)? Sometimes I see a command like  find . -name * -exec ls -a {} \;   I was asked to execute this.  What does {} \; mean here? Does anybody know which files I need to save in order to have a simple backup?   Reason for my question is, that my Galaxy Android smartphone lost all calendar data without any interaction. I found tons of apps, tons of sync stuff, but NO simple list of files to save. I do NOT want to save my data remotely on Google.  There must be at least one local file (database?) on my phone containing the synced calendar file (on my server, using ICSSync) Trying to understand formula for counting regions of hyperplane arrangements in $\mathbb{R}^2$ There are up to $\binom{n}{2} + n + 1$ regions created by a hyperplane arrangement in $\mathbb{R}^2$ containing $n$ hyperplanes.  I want to understand this in a demonstrative way. Each hyperplane splits each region it passes in two regions. A newly added hyperplane therefore creates a new region each time it crosses another hyplerplane or the plane. So the number of regions should be $1$ (plane) $+ \binom{n}{2}$ (number of intersections of $n$ lines). But as I know the answer, I know that I am missing $n$ regions.   Do you know where my mistake is? Why do I have to add the number of lines to get the max. number of regions? Got this problem from a friend and would like some pointers if anyone has any.  Fill an $n \times n$ matrix with the symbols $\{a,b,c,d\}$ subject to the following:     Only the symbol $a$ appears to the left or above an $a$    Only the symbol $b$ appears to the right or above a $b$    Only the symbol $c$ appears to the left or below a $c$    Only the symbol $d$ appears to the right or below a $d$     Here are some examples of a valid fillings: $$\left( \begin{matrix} a &amp; a &amp; b  &amp; b \\ a &amp; a &amp; b &amp; b \\ c &amp; c &amp; c &amp; d \\  c &amp; c &amp; c &amp; d  \end{matrix}\right) $$  $$\left( \begin{matrix} a &amp; a &amp; a  &amp; a &amp; a &amp; b  &amp; b \\ a &amp; a &amp; a  &amp; a &amp; a &amp; b  &amp; b \\ a &amp; a &amp; a &amp; d &amp; d &amp; d &amp; d \\  a &amp; a &amp; a &amp; d &amp; d &amp; d &amp; d\\a &amp; a &amp; d &amp; d &amp; d &amp; d &amp; d\\c &amp; c &amp; d &amp; d &amp; d &amp; d &amp; d\\  c &amp; c &amp; d &amp; d &amp; d &amp; d &amp; d  \end{matrix} \right) $$  What I have done: We'll it is clear to me that the $a$'s (for example) make up a sort of Young Diagram just by the property above. But there is more to the problem than that. If your YT for $a$ is just a square, then you might have a reversed (right justified) YT available to place for the $b$'s. Details like this seem to make this a tricky problem for me. I am intrigued by this problem and I just thought I'd share it here in case there is some easy way to do this counting problem that I've never heard of.  Thanks, Prove that any complex matrix can be expressed in term of Hermitian and anti-Hermitian matrices How can we prove that any complex matrix can be expressed in term of hermitian and anti-hermitian matrices of the following form $A = H_1 + iH_2$ Where, $H_1$ is hermitian and $H_2$ is an anti-hermitian matrix