I came across the following question:  Find the number of solutions of the equation $3^x+4^x=5^x$ in the set of positive real numbers.  I tried the above question by taking log on both sides and then solving, but it didn't seem to work. Any idea on how to proceed to solve this question?
I would like to prove that the equation $ 3^x+4^x=5^x $ has only one real solution ($x=2$)  I tried to study the function $ f(x)=5^x-4^x-3^x $ (in order to use the intermediate value theorem) but I am not able to find the sign of $ f&#39;(x)= \ln(5)\times5^x-\ln(4)\times4^x-\ln(3)\times3^x $ and I can't see any other method to solve this exercise...
I've got a 32-bit Ubuntu installation running on 64-bit hardware. Now that , I would like to switch to 64-bit without having to reinstall the OS.  This is one of the :  Shawn installed his system using the 32-bit version of Ubuntu, but his hardware is 64-bit and he wants to switch over. He manually installs the amd64 versions of dpkg and apt, replacing the i386 versions and changing which architecture is used as the default; then he installs the amd64 ubuntu-minimal package; then he installs the amd64 ubuntu-desktop package. Over time the remaining i386 packages are replaced automatically on upgrade.  However, when trying to follow the instructions in there, I cannot find any 64-bit version of dpkg or apt.  Did this user story got implemented in a different way in the final spec, or do I need to do something differently?  In short, how can I switch my 32-bit installation to 64-bit? Is is possible that a software issue could be causing spotty USB-C connections on 2017 MacBook Pro? I have a 2017 MacBook Pro. It has 2 USB-C ports on the left side and I use an Aukey hub to provide more ports (4 USB, 1 HDMI, 1 USB-C) and always to connect two monitors.   I connect one monitor through the hub with the HDMI port and another directly to the MBP with a USB-C adapter. Sometimes when I connect the hub to the Mac and the 2nd monitor to the Mac, it works as it should with no problem...but more than half the time I have to connect, disconnect, jiggle, jangle, switch ports, reconnect HDMI, etc. and hopefully they will both work. OFTEN, I can only get one monitor to fire up.  Is there any kind of software fix or reset that might solve this issue? Any other thoughts welcome. Thanks. Is "Five dollars is all I have" correct? There are no compact minimal surfaces This is one of the exercises of 'Do Carmo' (Section 3.5, 12)     How do you prove that there are no compact (i.e., bounded and closed in $\mathbb{R}^3$) minimal surfaces?   Thanks! How to read ext4 partitions in Windows? I have two questions on ext4:   Is there currently a way to mount an ext4 partition in Windows (XP/Vista/7) to get basic read support? If not, are there any plans to support this someday? How to automatically fit many bars in barchart using pgfplots? I am trying to use PGFPlots to display a large amount of bars, and I run into the problem that it cuts off the last of them.  MWE:  \documentclass[crop,tikz]{standalone}  \usepackage{pgfplots} \usepackage{pgfplotstable} \pgfplotsset{compat=1.13}  \begin{document}  \pgfplotstableread{ row label runtime 0 1 5264 1 2 2971 2 4 1866 3 8 1384 4 16 1205 5 32 1144 6 64 1109 7 128 1095 8 256 1108 9 512 1178 10 1024 1335 11 2048 1667 12 4096 1588 13 8192 2843 14 16384 5541 15 32768 10985 16 65536 21873 17 131072 43687 18 262144 87064 19 524288 174119 }\mytable   \begin{tikzpicture}   \begin{axis}[         ylabel={runtime (${\mu}s$)},         xticklabels from table={\mytable}{label},         xtick=data,         ymin=0,         ymax=10000,         ybar,         xticklabel style={           anchor=north west,           align=left,           rotate=-45,           font=\tiny,           inner sep=1pt,         }, ] \addplot+ [] table [y=runtime, x expr=\coordindex] {\mytable}; \end{axis} \end{tikzpicture}  \end{document}   Result:    By adding the following to the axis options, I can make all the bars fit. But I need to manually adjust bar width (and xmax)  xmin=-1, xmax=20, bar width=7pt,     Is there a way to handle many bars in a barplot automatically? Distorted Cabin & Oily Sheen: What is going on in this paragraph? I have been reading a book, and I have a question about the following excerpt:  &quot;We're here,&quot; The Tortoise in the Lake said, cutting across Dong Huong's gloomy thoughts. &quot;I can see nothing.&quot; There came a low rumble, which distorted the cabin around her, and cast an oily sheen on the walls. &quot;Watch.&quot; Outside, everything was dark. There was only the shadow of The Two Sisters in Exile, the dead ship that they'd been pulling since Longevity Station. – The Two Sisters in Exile, a fantasy novel by Aliette de Bodard  Are these changes happening in the “real” life of the novel?, or are they an illusion that exists merely in the mind of the narrator? If they are only an illusion, why does the author not say so? Has the author omitted something, or are these changes are really happening as the author says? I would very much like that the the appendix sections of the Table of Contents doesn't have a section numbers, dots, and page numbers. If I do the following  \documentclass{article} \usepackage{tocloft} \addtocontents{toc}{\cftpagenumbersoff{section}} \renewcommand{\cftdot}{} \begin{document} \tableofcontents \newpage \section{Appendix} \section*{Another Letter} \addcontentsline{toc}{section}{Another Letter} \end{document}   then the section "Another Letter" is left aligned, so it doesn't look like the  others.  Question  Is it possible to have "Another Letter" styled the same way as normal TOC entries, but just without section number, dots, and page number? Beamer, page/frame numbers, buttons and Appendix slides SQL Server 2012 and 2016 Standard:  If I put if-else logic in a stored procedure to execute one of two branches of code, depending on the value of a parameter, does the engine cache the latest version?  And if on the following execution, the value of the parameter changes, will it re-compile and re-cache the stored procedure, since a different branch of the code must be executed? (This query is quite expensive to compile.) What is the need of complex functions in wave analysis? It is commonly known that waves can be express in terms of sine or cosine function. But when I study further, I seen that for analyising the waves, it is common to use complex functions in the form $$y=y_{_0}e^{i(kx-\omega{t})}$$ where $y_{_0}$ is the amplitude, $k$ is the wave number, $\omega$ is the angular velocity and $x$ &amp; $t$ are position an time respectively. Ofcourse, I know that the function $e^{ix}$ can be written in the form $\cos{x}+i\sin{x}$ and so it is a periodic function with period $2\pi$ but my question is for what purpose we define it in terms of complex numbers? It seem to be more convenient to use real functions for real variables such as amplitude, electric and magnetic field of an electro magnetic wave, and also in quantum mechanics. What actually this interpretation means or what is the advantage of such functions? How can I wrap text around a bottom-right div? Every time I try to do something seemingly-simple in CSS, it doesn't work.  I have a content div that contains a 460x160 image. All I want to do is position the image at the bottom-right corner and wrap my text around it.  &lt;div id="contents"&gt;     &lt;img src="..." /&gt;     text text text text text text ... &lt;/div&gt;   So I want it to look like:  ------------------ | text text text | | text text text | | text text -----| | text text |    | ------------------   Doing it with a top-left or top-right image is cake:  #contents img { float:right; }  ------------------ | text text |    | | text text -----| | text text text | | text text text | ------------------   Now how do I push that to the bottom? The best I've come up with so far are:  #contents img { float:right; margin-top: 90%} // really needs to be 100%-160px  ------------------ | text text      | | text text      | | text text -----| | text text |    | ------------------   In this case the text does not print in the margin, so there is white space above the image.  #contents { position:relative; } #contents img { position:absolute; right:0; bottom:0; }  -or-  // move the img tag under the text in the html and: #contents img { float:right; margin-top:-160; }  ------------------ | text text text | | text text text | | text text -----| | text text | te | ------------------   In this case the text prints over or under the image.  So... how can I accomplish this? What motivates people to answer questions in Stack Overflow? I want that too - motivation to answer programming-related questions. What is yours? What is the difference between  and != Fast(est) and intuitive ways to look at matrix multiplication? Play Framework (2.1.3) doesn't run any tests STM32 & ST-LINK - Cannot connect to MCU after successful programming I have built my own board with STM32F7-45VGT6. I have successfully programmed it with ST-LINK v2 (not the original one though) and now I cannot even connect with MCU.  I use ST-Link Utility from ST and SWD interface. It can be the case that I use SWD pins as output and in my code I set them as GPIO output. Can it be the case?  Nevertheless, I connect my reset pin to GND and set "Connect under reset" option in ST-Link Utility but it doesn't work... What can I do?  On the Internet, I have found something about using BOOT0 Pin, but I don't know exactly... Let $P\in\mathbb{R}[X]$ be a positive polynomial.  I want to show that there exists $A,B\in\mathbb{R}[X]$ so that $P=A^2+B^2$    $\displaystyle P=a\prod_{k=0}^q(x-a_k)\prod_{k=0}^{2p}(x-b_k)=a\prod_{k=0}^q(x-a_k)\prod_{k=0}^p(x^2-2\Re(b_k)+|b_k|^2)$ where $a$ is the leading coefficient, $a_k$ are the real rooks, $b_k$ are the complex roots.  What can i do now ? I know that the identity $(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$ should come in handy, but I can't figure out :  1) How to show that the product real roots is a square  2) How to use the above identity for the complex roots.  If I can solve those two points, then I can solve the problem. I have been looking at this for hours and it isn't making anymore sense than it did in the first hour.   If $a$ und ${x_{0}}$ are positive real numbers and ${x_{k}}$ defined as follows, prove that ${x_{k}}$ is monotone decreasing and bounded, then calculate the limit.  ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$  What I though I had to do was pick an ${x_{0}}$ and solve for ${x_{k}}$, so I picked ${x_{0}}$. Then I wanted to put the result back into the function to get ${x_{k+1}}$, which I still believe is what I'm supposed to be doing, but I don't understand what I am getting as a result.  I get that I should prove it is decreasing, then that it is bounded,  then address the limit, but the how is missing. I wonder if it is possible to consider $\mathbb R$ as a Cartesian product $X\times X$ for some set $X$. From the point of view of the dimensionality, there are spaces with a Hausdorff dimension $1/2$ (sort of Cantor sets), but I guess there are other problems in this construction  unrelated to the dimension.  Edited: replying on the comment by Asaf. I want that for all $r\in\mathbb R$ there exists a unique representation $r=\langle x,y\rangle$ where $x\in X$ and $y\in X$. Also, what if we want to have $X$ to be a topological space and $h:X\times X\to\mathbb R$ to be a homeomorphism?