Find all continous function satisfying: $\int_{0}^{x} f = f^2(x) + C$, for some constant $C$ This is problem 7 of Chapter 14, Calculus by Spivak. My approach involves differentiating both sides of the equation. But by doing so, we assume the differentiability of  $f$ in $R$. It's impossible to think of something else other than considering derivatives. So let we say: $f$ is a differentiable function for simplicity. So we have: $$\int_{0}^{x}f = f^2(x) +C \Leftrightarrow f(x) = 2f(x)f'(x) \Leftrightarrow  \left[  \begin{array}{rcr}     f(x) =0 \\     f'(x) = \frac{1}{2} \\   \end{array}(**) \right.$$ There are two cases to consider: either $f(x)$ is $0$ for every x in R or there exists some $x$ satisfying $f(x) \not= 0$ So, $f(x) =0,\forall x\in R$ is one solution. The other solutions are $f$(s) such that: $f'(x) = \frac{1}{2}$, for $x$ such that $f(x) \not= 0 $ It seems that there's no guarantee that the set $\{x: f(x) \not= 0\}$ can be any specific set. If we assume $f(x) = 0$ only once at $0$ then $f(x) = \frac{x}{2}$ is another solution. But is that all in the second case? I'm trying to somehow exclude other cases. I haven't found any promising approach yet. Any ideas ?
Find all continuous functions $f$ satisfying   $$\int_0^xf=(f(x))^2+C$$ for some constant $C \neq 0$, assuming that $f$ has at most one $0$.  I have a question about the solution to this problem. It says that   Clearly $f^2$ is differentiable everywhere, its derivative at $x$ is $f(x)$ (I'm assuming that this is because the left hand side is differentiable, the right hand side must be differentiable as well). So $f$ is differentiable at $x$ whenever $f(x) \neq 0$, and  $$f(x)=2f(x)f'(x),$$ so $f'(x) = \frac{1}{2}$ at such points. Thus, $f= \frac{1}{2}x+b$ for some $b$ on any interval where it is non-zero; if $f$ has a zero, this gives two possible solutions, with two possible values of $b$, but since $f$ is assumed continuous, they must be the same. So we need $$\int_0^x(\frac{1}{2}t+b)dt=(\frac{1}{2}x+b)^2+C$$  so we must have $b=\sqrt{-C}$ or $-\sqrt{-C}$, leading to two solutions for $C \lt 0$.  I don't understand the bolded statements. First, how does the differentiability of $f^2$ guarantee that $f$ is differentiable, and why only at $x$ where $f(x) \neq 0$?  Next, why are there two possible solutions if $f$ has a zero, but since $f$ is continuous they must be the same?  Finally, does the last statement mean that if $C \gt 0$, then there is no solution, since we're restricted to real numbers here?  I would appreciate it if anyone answers my questions.
datatool with longtable Is it possible to schedule my laptop to mute and unmute at a specific time? I always used to wake up to my favorite radio station on my alarm clock. However, I have moved and can no longer get that radio station over the air. To solve this problem, I would like to stream the station on my laptop, but have it muted overnight, then set it to unmute at a specific time in the morning, so that it will act as an alarm clock. Is there a way to schedule my laptop (Windows 7 64-bit) to unmute at a specific time? Google Docs thumbnails do not update, can I force an update? ERROR running force:source:pull: Entity of type 'CustomApplication' named 'Contact' cannot be found How to check if a binary is 16 bit on Windows Multiple TikZ keys with foreach Prove that there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$. Prove that there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$.   This is a homework question, and I understand that its part of Fermat's Last Theorem, but when I looked that up to try to figure out the homework, I realized that it is way further than what we've learned in class so far. We have to prove this, but the only way I can think to do it is by exhaustion, which would be extremely lengthy. Any little shove in the right direction would be appreciated. Thanks! What good are free groups? In , one learns two definitions of free groups associating with sets.        Let $A$ be a set, the free group of $A$, $F(A)$ is the initial object in the category $\mathcal{C}$, where \begin{equation} \operatorname{Obj}(\mathcal{C})=\{A\xrightarrow{g}G\}, \end{equation}where the codomain $G$ are groups, and \begin{equation} \operatorname{Hom}(\mathcal{C})=\{\text{Commutative Diagrams } A\xrightarrow{g_1}G_1\xrightarrow{\phi}G_2\xleftarrow{g_2}A\}  \end{equation} where $\phi$ are group homomorphisms.   Also $F(A)$ has the concrete construction with elements being non-redundant words with alphabet $A$, and the group multiplication being juxtaposition and reduction.      But what good are free groups? Why are they useful?  says they are useful in topology, but does not explain why explicity.   Can someone give some examples that an undergraduate can understand? I am asking about examples that can show the usefulness of this abstract construction. So I guess the identification of $\mathbb{Z}=F({a})$ does not really count. Should a toggle button show its current state or the state to which it will change? I own a Panasonic DMC-G81 (same as G80/G85) with in-body Image Stabilization.  When turned on everything is just fine, but when turned off, the sensor is wobbling around in the body. When no lens attached, it's clearly visible moving about 1/2 cm in the body. I didn't find anything about it in the internet. Shouldn't the sensor be in a kind of parking position, should I worry about the sensor ?  I'm not sure if that is the usual behavior, maybe someone with the same or similar body can have a look. has been, have been , had been Please anyone tell me when we have to use "has been, have been , had been" and something  started in past and still continuing is it possible to use has been ? show that $$\lim_{n\to \infty}\sin{(n^m)}$$  divergent,where $m\in N^{+}$  I have kown $$\lim_{n\to\infty}\sin{n^2}$$  to be divergent and dense in $[-1,1]$. This is very  famous problem,the problem is first is post $AMM$( The American Mathematical Monthly,1970-1975) problem,and after some years  $AMM$ post $\sin{(n^{14})}$ in $[-1,1]$ was dense? and this is open problem.so I Guess  $$\sin{(n^m)}$$ is dense in $[-1,1]$. App Store is stuck in a loop trying to download a free app -- how do I make it stop? My App Store keeps saying this item is free in the store, that this item will be added to your purchases and it then says cancel or download.   I click download and it won't download and keeps popping up the same message over and over again.  How do I make it stop? I can't download any apps. Addons: Only some .zips will install My display driver's maximum screen width is 1280 pixels.   I need to test a website with resolution 1920&times;1080 pixels. Or 3000&times;5000.   Different browsers is optional. I'm trying to delete some old flow which are inactive via workbench. DestructiveChanges.xml &lt;?xml version=&quot;1.0&quot; encoding=&quot;UTF-8&quot;?&gt; &lt;Package xmlns=&quot;http://soap.sforce.com/2006/04/metadata&quot;&gt;     &lt;types&gt;         &lt;members&gt;Advanced_Round_Robin&lt;/members&gt;         &lt;name&gt;Flow&lt;/name&gt;     &lt;/types&gt;Ï &lt;/Package&gt;  package.xml &lt;?xml version=&quot;1.0&quot; encoding=&quot;UTF-8&quot;?&gt; &lt;Package xmlns=&quot;http://soap.sforce.com/2006/04/metadata&quot;&gt;       &lt;version&gt;48.0&lt;/version&gt; &lt;/Package&gt;  I've added both file in zip and in workbench when I deploy using this zip file I get below error.  insufficient access rights on cross-reference id   What permission I need to have to perform this action? Also is there way I can delete few version of same flow only? Like keep only last 3-4 version and delete all previous version of given flow? Thank you. How to pass parameters to query? Is casting the same thing as converting? In Jesse Liberty's Learning C# book, he says "Objects of one type can be converted into objects of another type. This is called casting."  If you investigate the IL generated from the code below, you can clearly see that the casted assignment isn't doing the same thing as the converted assignment. In the former, you can see the boxing/unboxing occurring; in the latter you can see a call to a convert method.   I know in the end it may be just a silly  semantic difference--but is casting just another word for converting. I don't mean to be snarky, but I'm not interested in anyone's gut feeling on this--opinions don't count here! Can anyone point to a definitive reference that confirms or denies if casting and converting are the same thing?      object x;     int y;      x = 4;      y = ( int )x;      y = Convert.ToInt32( x );   Thank you  rp  Note added after Matt's comment about explicit/implicit:  I don't think implicit/explicit is the difference. In the code I posted, the change is explicit in both cases. An implicit conversion is what occurs when you assign a short to an int.  Note to Sklivvz:  I wanted confirmation that my suspicion of the looseness of Jesse Liberty's (otherwise usually lucid and clear) language was correct. I thought that Jesse Liberty was being a little loose with his language. I understand that casting is routed in object hierarchy--i.e., you can't cast from an integer to a string but you could cast from custom exception derived from System.Exception to a System.Exception.   It's interesting, though, that when you do try to cast from an int to a string the compiler tells you that it couldn't "convert" the value. Maybe Jesse is more correct than I thought! What methods of suicide cause the least amounts of collateral damage? CK2 is a big game with lots of variables, so I'll stick to adding them upon being requested to minimize the hassle.  Anyway, let's say that you have a lot of children and the oldest one claims all the land according to laws of succession. After the father dies, the circumstances will obviously piss off a lot of people (in my case that would be 6 out of 8 vassals and 4 siblings with an opinion of -50 and below one of which is a sister married to the Basileus of the Byzantine friggin' Empire, a case of excommunication, 4 failed assassination attempts under 3 years... and lots and lots of vikings). Let's also say that the current playable character is completely incompetent compare to the next sibling in line.  So, what would be the most optimal methods to off myself without further screwing over the countryside so the next heir in line has to fix it (if we assume that said heir is actually capable of doing so)?  This is my list of DLCs (not including music and unit packs):   Charlemange Legacy of Rome Sword of Islam The Republic Way of Life Number of vertices present in all maximum matchings Given a graph $G$, we need to find the cardinality of the largest set of vertices so that each of them are present in every maximum matching possible.   Is there a solution beside the obvious remove each vertex and find the maximum matching to see it reduces?