This problem is quite easily solved by using logarithms and derivative and forming the function $f(x,y) = x^y - y^x$, however there are assertions that this problem can be solved without using either of the two. I can not see how one would proceed to solve this problem in the absence of logarithmic manipulation and differentiation. Can anyone shed some light on this?
I've been investigating the Cartesian graph of $x^y=y^x$. Obviously, part of the graph is comprised of the line $y=x$ but there is also a curve that is symmetrical about the line $y=x$. (We can prove this symmetry by noting that the function $x^y=y^x$ is self-inverse; all self-inverse functions are symmetrical about the line $y=x$.) An image of the graph is shown below:  I decided to find the intersection point and arrived at an intriguing result: the intersection point between the two curves is at $(e,e)$. The following is my method: If the gradient of the line $y=x$ is $1$, the gradient of the curve at the intersection point must be $-1$ as it's a normal to the line (as it's symmetrical about the line). This means that at that point $\frac{dy}{dx}=-1$. Now to find $\frac{dy}{dx}$. We have $x^y=y^x$. I then used a very powerful tehnique for differentiating these sorts of functions. We know that eg $$x^y=e^{\ln{x^y}}=e^{y\ln{x}}$$ and $$\frac{d}{dx}e^{f(x)}=f'(x)e^{f(x)}$$ Applying it to our function $x^y=y^x$ and using implicit differentiation and the product rule gives us: $$(\frac{dy}{dx}\times \ln x +\frac{y}{x})x^y=(\ln y +\frac{dy}{dx}\times \frac{x}{y})y^x$$ So $$\frac{dy}{dx}x^y\ln x +yx^{y-1}=\frac{dy}{dx}xy^{x-1}+y^x \ln y$$ Extensively rearranging gives: $$\frac{dy}{dx}=\frac{y^x \ln y -yx^{y-1}}{x^y \ln x -xy^{x-1}}$$ Let $\frac{dy}{dx}=-1$: $$y^x \ln y -yx^{y-1}=xy^{x-1}-x^y\ln x$$ But we know $x=y$ since we are at the intersection point with the line $y=x$: $$x^x \ln x -x^x=x^x-x^x\ln x$$ So $2x^x \ln x -2x^x=0$ $$2x^x(\ln x -1)=0$$ This means either $2x^x=0$ or $\ln x -1=0$ but we know $x^x$ is always greater than $0$ so $\ln x =1$, leaving us with: $$x=y=e$$ So I have my answer, but is there any other method of getting it? I have heard there is. Any help will be very welcome. Thanks in advance
Simple chessboard exercise What alternative notation of subscripts/superscripts that are also treated as SYMBOLS in Mathematica? How to calculate the rotation matrix between 2 3D triangles? I need to calculate the rotation matrix and the translation vector between 2 given triangles in Euclidean space.  This is really just a rotation and a translation, the lengths of sides of the triangle are equal.  Coordinates of the points are real. $P_1, P_2, P_3$ are the 3 points of the 1st triangle, $Q_1, Q_2, Q_3$ are the 3 points of the 2nd.  My idea was: I translate the center of the triangles to the point of origin with $t_1 := \frac{P_1+P_2+P_3}{3}$, $P'_i = P_i-t_1$ and  $t_2 := \frac{Q_1+Q_2+Q_3}{3}$, $Q'_i = Q_i-t_1$ and then I only need to solve the set of 9 linear equations with 9 unknown (the 9 entries of the rotation matrix $R$):  $$ Q'_i = R.P'_i $$ Afterwards I could calculate the translation with  $$ t = t_2 - R.t_1 $$ The problem is: after I translate the triangles to the point of origin the points are linearly dependent ($P_3 = -P_1 - P_2$)  So in fact I only have 6 equation with 9 unknown, which allows multiple solutions for $R$. But I need "one" precise solution, because the triangles are just parts of two identical models, which I need the rotation for.   I think the approach will work, if I use 4 points instead of 3.  But can I somehow calculate the rotation matrix with "only" 3 points? How can I access the apache basic-auth username in a PHP script? Find all solutions of the equality $y^2=x^3+23$ for integers $x,y$   I guess, that x cannot be even. because we can apply mod 4 test  say $x=2k$ then $y^2=8k^3+23\equiv3\mod(4)$   but, this is not possible, since the square of an integer is congruent to $0$ or $1$ $\mod (4)$  so $x$ is of the form $4k+3$ or $4k+1$  $x=4k+3$ is also not possible for the same reason but for the last case the test fails.  any hints ? When should I add another answer? I have been member of SO for 4 months. I have a question regarding some functionality. Whenever I give an answer to a question, there is functionality given Add Another Answer. I can comment on my answer at any time and also able to edit answer. Then   Why this functionality given on SO to add another answer?  When should I add another answer?  Why there is requirement of multiple answers? What is limit of adding answers to single question? For an existing set of line features, I would like to generate a perpendicular line, of say 1km, at the midpoint of each existing line feature in ArcGIS.  I imagine that The methodology would be to:   create a tangent line at the midpoint of each existing line feature calculate the angle of the tangent line based on the angle of the tangent line, calculate the location of the endpoint of the new line create the new line with the calculated endpoint of the new line and the existing midpoint of the old line   Let's assume that we are in a UTM projection and can use coordinates to calculate distance and direction.  How would I do this with ArcGIS tools or with ArcPy? What's a good Windows RDP client for Mac? I administer a number of Windows machines, and am trying to find a really good application to remote desktop in to them from my Mac.  I've tried CoRD, and I really like its library approach to saved sessions (as opposed to Microsoft's file based approach), and tabbed sessions are great when working on a bunch of machines.  Unfortunately, it's just too buggy for serious use, and given that it averages less than 2 releases a year, I have little hope that that will change.  Microsoft's official client is very bare bones and doesn't support connecting on non-standard ports (which I need).  Is there a remote desktop client for Mac out there that's both fully featured and not buggy?  Edit: Apparently the Microsoft doc I read that states that the Microsoft RDC client can't be used on non-standard ports is out of date. Hello can someone give me short tutorial how to:   Remove Windows 7 from my dual boot system (Ubuntu/Windows7) Let me use free space as new partition for data (so I don't need to extend Ubuntu itself here) Remove windows 7 option from boot screen   Update:  Thanks to Lekensteyn answer I successfully deleted windows. Now I created new partition (formatted to ext4), but I can not use it. I cannot copy files there, only as root.  Also when booting, Ubuntu complains it didn't find /windows partition (of course it didn't, I removed it). I feel like I should update some tables or what, any ideas? What is the difference between '/' and '//' when used for division? Is there a benefit to using one over the other? In Python 2, they both seem to return the same results:  &gt;&gt;&gt; 6/3 2 &gt;&gt;&gt; 6//3 2 Is it possible to see the history of changes to a particular line of code in a Subversion repository?  I'd like, for instance, to be able to see when a particular statement was added or when that statement was changed, even if its line number is not the same any more. How can I get file extensions with JavaScript? Question : $$\begin{align}a+b+c &amp;= 0\\ a^3 +b^3 +c^3 &amp;= 12\\ a^5 +b^5 +c^5 &amp;= 40\end{align}$$ Then , $a^4 + b^4 + c^4 = ?$  My try : As a common perspective I just went to find any trick related to it and my first step is as usual as a common man will think $3abc = 12$ and got $abc = 4$ After that I am unable to link it with any other equation Why is my Spring @Autowired field null? I want to describe the variables involved in an equation in an itemized environment such that every item begins with the variable name and subsequently comes the description for it. I'd like to have all the descriptions left aligned.  Here is a MWE that of course does not produce what I am looking for.  \documentclass{standalone} \begin{document} $E[g(x)] = \int_{-\infty}^{+\infty} f_X(x) \, g(x) \, \mathrm{d}x$. % where, % \begin{itemize}     \item $x$      \quad is a continuous random variable.     \item $f_X(x)$ \quad is the PDF of $x$.     \item $g(x)$   \quad is a function of $x$. \end{itemize} \end{document} How to allocate aligned memory only using the standard library? I just finished a test as part of a job interview, and one question stumped me, even using Google for reference. I'd like to see what the StackOverflow crew can do with it:  The memset_16aligned function requires a 16-byte aligned pointer passed to it, or it will crash. a) How would you allocate 1024 bytes of memory, and align it to a 16 byte boundary? b) Free the memory after the memset_16aligned has executed.  {        void *mem;    void *ptr;     // answer a) here     memset_16aligned(ptr, 0, 1024);     // answer b) here     } A method for the OP to accept comments? Question: What if the author of a question can accept a comment?   The problem:  Time and time again, answers are posted as a comment.  Usually, the person who commented has the best of intention (e.g. it was a trivial answer [to the answerer], and they weren't sure if the OP would immediately edit the question after the comment).  But this leaves the frustrating situation of an unanswered question with a perfectly good answer.  This problem accumulates over the years.     Candidate solution:  The OP can just "accept" the comment, after which a duplicate of the comment is made an accepted CW answer, with a preface indicating that it was originally a comment, e.g. This answer was originally a comment posted by [[blah]] that has been accepted by the author of the question.   Comments on this scheme:   The author of the comment has indicated they're not interested in gaining/losing reputation through their comment (thus CW answer). The original comment is still there. If desirable, the "author" of the accepted question could be a "bot", so the answer is not attributed to a genuine account.  (I suppose it would already be technically possible for Joe Blow to create a script for a bot that creates an answer from a comment in the proposed way.)     A similar question was asked before  (over 3 years ago), and in numerous other questions which have been deemed "duplicates" (,  and ; see also ), so there seems to be demand for such a feature.  The answers to these questions seem (to me, at least) quite weak:   "ask the commenter to add the comment as an answer"   The OP already did this by asking the question in the first place.  And what if the author of the comment doesn't respond (or takes a long time to respond)?  This also adds more chit-chat irrelevant to the question.   "add an answer of your own, referring to the comment, and accept it."   A hacky workaround.  How could this possibly be better than accepting a comment?  (Yet, this is the most up-voted response.)  Moreover, this adds to the user's list of "answers" (thus making the user's page harder to look through).   "When i leave a comment, i do so because i want to leave a comment."   Great; now what about everyone else?  In the immortal words of the FAQ: "Other people can edit my posts?!"  And it's not even an edit, it's copying the post from one box to another (a more hacky version of which is regarded as the "accepted" method). Number of equivalence relations on a set What is a terminal and how do I open and use it? Do I need a Turkey transit visa for a layover? I have an Indian passport and am planning to visit Germany with a valid visa. To book a flight for Germany (Mumbai – Dusseldorf), I came across some connected flights that stop at Istanbul.  Do I need to have visa for Turkey?  This is the first time I am travelling outside my country, and any information on this will be helpful.