Let define $f:\mathbb{R}\rightarrow\mathbb{R}$ as $f(x)=3^x+4^x-5^x$.  Prove that there exists only one $x_0$ such as $f(x_0)=0$.  My approach:  We can see that $\lim_{x\rightarrow-\infty}f(x)=0$ and $f(2)=0$. I don't know how to use the derivatives of $f$ to prove that it is firstly increasing and then decreasing for $x\in(-\infty,2)$ and for $x\in(2,\infty)$ it is only decreasing.
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...
Passing an enum value as command parameter from XAML I want to pass an enum value as command parameter in WPF, using something like this:  &lt;Button      x:Name="uxSearchButton"      Command="{Binding Path=SearchMembersCommand}"      CommandParameter="SearchPageType.First"     Content="Search"&gt; &lt;/Button&gt;   SearchPageType is an enum and this is to know from which button search command is invoked.  Is this possible in WPF, or how can you pass an enum value as command parameter? Can every game of Klondike-Solitaire be solved? I play a lot of Solitaire games on my Android phone and love to keep an eye out for the statistics.  Given that the Solitaire version lets you restart the game endlessly, I usually play until I solve it. But I never managed to solve more than 80% of the games played (1000+).  So now I wonder, is every Solitaire game solvable? Typesetting linguistic rewrite rule I'm trying to (more or less) replicate this phonological rewrite rule:    I'm using the package , and what I manage to accomplish is this:  \documentclass{article} \usepackage{phonrule} \begin{document}  \phonc     {\phonfeat{$-$cnt\\ +hi}}     {[$\alpha$grv]}     {\oneof{%         \phonfeat{$-$cnt\\ $\alpha$grv}\phold\\         \phold\phonfeat{$-$cnt\\ $\alpha$vce}%     }}  \end{document}     My main problem here is what appears after the "when" slash:  (1) The package only provides a left curly bracket - how can I add a right one as well?  (2) The two options within the curly brackets are not well aligned. I think comparing the two pictures themselves says it better than what I could do with words. Where can I rent a decent mountain bike in Zürich? I am going for a 4 day trip in Zürich and I would like to bike quite a lot. Though biking is quite advertised, I failed to find a decent bike rental shop in the city. Would highly appreciate some tips! Applications Not Reopening After Restart Transit visa requirements at Narita Airport How to type en-dash and em-dash in Windows 7/8? I write blog posts, and I often need to type en-dash (–) and sometimes em-dash (—) in my blog posts. Currently I keep these characters saved in a text file, and copy-paste them whenever I need them. Unnecessary to say, it is a headache to move between keyboard and mouse when you type large texts.  It would be better for me if there ware a keyboard shortcuts available for these two characters. Is there a way to type en-dash and em-dash in Windows? I use Windows 7 and Windows 8 in different computers.    Here's an aligned comparison of dashes, if you need to see the difference for some reason.   Here's-dash Here's–en-dash Here's—em-dash How do you normalize a zero vector How to prove polynomials with degree $n$ does not form a vector space? This is one of my linear algebra problems:     Prove that polynomials of degree $n$ does not (The professor made these words bold intentionally) form a vector space.   From what I read, the set of polynomials of degree $n$ should be a vector space, because:   There is an "One" and a "Zero" in this set; We can find inverse for addition and multiplication from this set; It follows all the axioms of addition. It follows all the axioms of scalar multiplication.   Then can someone give me some hints to prove it does not form a vector space? How can I explain to people that the phrase “off of ” is grammatically incorrect? How can I explain to people that the phrase off of is grammatically incorrect?  I‘ve heard this phrase used a lot, especially by Americans (though they aren't the only ones).    In my understanding, off of should usually be replaced by off, as in,     I took the book off the shelf   as opposed to     I took the book off of the shelf   Am I wrong?  Or is there perhaps some simple way I can explain this to most people? Are nuclear weapons simply non-existent in the Star Wars universe? The resistance is attacks the Starkiller Base with what appears to be pretty small time ammunition even by modern earth missiles, which requires a long running sequence of repeated flybys.  The objective, IIRC was to destroy a critical piece of machinery on the surface which would then destabilize the entire planet/weapon (feel free to correct in edit if I remember incorrectly).  OK cool, but then why the force wouldn't they just drop a nuclear bomb on it?  Have nuclear weapons been used any other time in Star Wars (including EU), or, like smart phones, have they just not been invented in that world? How do per-site Metas differ from Meta Stack Exchange, and where should I go to ask my question? How do per-site metas, such as Meta Writers Stack Exchange or Meta Stack Overflow differ from the general Meta Stack Exchange?  Where should I go to ask my question? Uniform convergence of derivatives, Tao 14.2.7. One click shutdown Ubuntu and load into alternative bootup CLIPBRD_E_CANT_OPEN error when setting the Clipboard from .NET Why does the following code sometimes causes an Exception with the contents "CLIPBRD_E_CANT_OPEN":  Clipboard.SetText(str);   This usually occurs the first time the Clipboard is used in the application and not after that. Is Python/ArcPy capable of delimiting strings with commas into different fields, and running a calculation based on those separate columns Using ArcGIS ArcMap 10.6: I have a text field in an attribute table that has comma separated values (example: 12,24, 64, 87,29). For this example, I will name this field VALUES. Please note that VALUES includes data collected from field work so the spacing between commas will be varied. My current workflow is as followed:  Export the ID and VALUES fields in the attribute table into a .csv file using Table to Table in ArcMap 10.6 Open the .csv in Excel and run the 'text to columns' function under the Data ribbon Separate the VALUES field into multiple columns based on commas Run the calculation: Copy Column I values and ID columns into a new .csv file Save and close the new .csv file and join the shapefile to the new .csv file based on ID Export the newly joined shapefile into a new file and delete the duplicate ID field in the new shapefile  I am wondering if there is a way to automate this Python/ArcPy so I can create a custom script tool to do this without leaving the ArcGIS environment. Is there a way to separate values in a text field based on commas into different columns, run the above calculation, and have the values from that calculation entered in a new field named VALUES_CALCULATED? Is there a fast way to get to the bottom of the Death Star? In Tiny Death Star, you can get to the top of your tower by tapping on the top of the screen while avoiding the "Add Level" button. However, the only way I can find to go to the bottom of the tower is to wait for someone to want an elevator ride, and tap on the button to give them that ride. Is there any other hot-spot for getting to the bottom of your Death Star? I have just downloaded "ubuntu-14.04.1-desktop-amd64.iso" and "ubuntu-14.04.1-server-amd64.iso" through "".   Please can someone confirm the maximum RAM that is supported.  According to many sites, Maximum RAM supported by the OS  ubuntu-14.04.1-desktop : " 2^64 bytes " -- Kindly confirm  ubuntu-14.04.1-server : " 2^64 bytes " -- Kindly confirm Voltage at soldering iron tip, should I be worried? I discovered that my soldering iron has voltage at the tip and it seems to increase the hotter the iron gets. Should I be worried? I've been trying to repair an old CRT monitor and I'm worried the issues I've been having are due to this cheap iron damaging something.  How to meassure voltage of an soldering iron tip? Where to put probes? How to remove zsh alias? I have an alias which is predefined by an oh-my-zsh plugin.  -&gt; % alias gcm gcm='git checkout master'  I'd like to remove it, i.e. I'd like alias | pcregrep "\bgcm\b" return 1. I've tried with alias gcm='' but after that the alias is still existent.