$f(0)=0$,$f(1)=1$,find the largest $u$ such that $\int_0^1|f'(x)-f(x)|dx \geqslant u$. Suppose   $E=\{f\in C^1[0,1]$:  $f(0)=0$,$f(1)=1\}$.   Find the maximal $u\in \mathbb{R}$,such that forall $f\in E$,   \begin{align*} \int_0^1|f'(x)-f(x)|dx \geqslant  u. \end{align*}   I can prove that $u\geqslant \frac1e$,since \begin{align*} \frac1e=e^{-x}f(x)\Big|_0^1=\int_0^1e^{-x}(f'(x)-f(x))dx\leqslant \int_0^1|f'(x)-f(x)|dx . \end{align*} But I don't know how to find the  maximal $u$.
Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$. Let $C$ be the class of all real-valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1$. How to show that $$\inf_{f\in C}\int_0^1|f'(x)-f(x)|dx=\frac{1}{e}?$$  I have been able to show that $1/e$ is a lower bound. Indeed, $$\begin{align*} \int_0^1|f'(x)-f(x)|dx &amp;= \int_0^1|f'(x)e^{-x}-f(x)e^{-x}|e^xdx \\ &amp;\geq \int_0^1\left(f'(x)e^{-x}-f(x)e^{-x}\right) dx \\ &amp;= \int_0^1 \frac{d\left(f(x)e^{-x}\right)}{dx}dx  \\ &amp;= f(1)e^{-1}-f(0)e^{0}\\ &amp;=\frac{1}{e}. \end{align*}$$ But how to show this is the infimum? Is there a function $f\in C$ such that we get $\int_0^1|f'(x)-f(x)|dx=1/e$?
NSArray becomes NSCFArray when passed app engine datastore auto-clears every time project runs Retrieving HTTP status code from loaded iframe with Javascript Applying for a UK Tourist Visa from outside country of origin How to know (in GCC) when given macro/preprocessor symbol gets declared? Suppose I have #define foo in various header files. It may expand to some different things. I would like to know (when compiling a .cc file) when a #define is encountered, to what it will expand, it which file it is and where it got included from.  Is it possible? If not, are there any partial solutions that may help?  Feel free to add comments with clarification requests.  Edit: current answers seem to concentrate on the case when there is one #define and I just want to jump to definition or know what the definition is. That's the simple case and yes, your solutions work. But when I have the same #define in different files, and want to know which one kicks in first, none of these techniques is useful. Okay, I actually used #warning carefully to find the right place. But this requires much work. I am trying to install Wordpress on Lubuntu. I followed the instructions . Including PHP Apache and MySQL setups. Apache and MySQL appear to be running fine.  However, when I go to http://localhost/blog/, I get the following error:     Neither /etc/wordpress/config-localhost.php nor /etc/wordpress/config-localhost.php could be found.   Ensure one of them exists, is readable by the webserver and contains the right password/username. Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$? Word for someone who acts like an expert but who has very little knowledge? What is a word to describe an individual who acts like an expert in a subject area, constantly stating facts and correcting people, but who actually has very little knowledge on the subject? My MATLAB launcher doesn't work I've installed MATLAB R2011a on Ubuntu 11.10 today following . I've also followed the steps to create the launcher. It appeared, but when I click on it nothing happens. Does anyone knows what can I do to fix it? Additional info I can run it perfectly through terminal. My only problem is getting the launcher to work. To create the launcher I used :   Get an icon:  sudo wget http://upload.wikimedia.org/wikipedia/commons/2/21/Matlab_Logo.png -O /usr/share/icons/matlab.png   Get the launcher file:  sudo wget 'https://help.ubuntu.com/community/MATLAB?action=AttachFile&amp;do=get&amp;target=matlab-r2011a.desktop' -O /usr/share/applications/matlab.desktop Convert a character digit to the corresponding integer in C When should we consider a dataset as imbalanced? I'm facing a situation where the numbers of positive and negative examples in a dataset are imbalanced.  My question is, are there any rules of thumb that tell us when we should subsample the large category in order to force some kind of balancing in the dataset.  Examples:   If the number of positive examples is 1,000 and the number of negative examples is 10,000, should I go for training my classifier on the full dataset or I should subsample the negative examples? The same question for 1,000 positive example and 100,000 negative. The same question for 10,000 positive and 1,000 negative. etc... How far does a rogue move when dashing twice using Cunning Action? A rogue receives a bonus Cunning Action, and one option is to Dash. My understanding is a Dash allows a player to move up to twice his speed for his turn.  If a rogue is using a regular action to Dash (hence, is already dashing) can he also use his cunning action as a bonus to Dash again — effectively moving 4 times normal speed in a turn? How to reverse a user input without using array or any library function(any function for reversing)? Given two sets$$A=\{1,3,5\}\quad,\quad B=\{1,3,8\}.$$ Then I compute the $A\setminus B=\{5\}.$   But my book$^\dagger$said: ... . Also, observe that $x\notin A\setminus B$ does not mean that $x\notin A\lor x\in B$. Why?  I don't know how to explain the Why, please give me some example.    $^\dagger$, second edition, page $4$.  EDIT: My intuition tells me that it does mean that. I guess there is some special case about null-set or about the restriction of the universal set. Sequence of a product of functions in $L^p. L^q$ with $p,q$ conjugate The question is: if $f_i$ is a sequence of functions in $L^p$ converging to $f$ and $g_i$ a sequence in $L^q$ converging to $g$ show that $f_ig_i$ converges to $fg$ in $L^1$ for $p,q$ finite and $\frac{1}{p}+\frac{1}{q}=1$. Does this result hold if $p=1, q=\infty$?  So I think I showed the first part. $||f_ig_i-fg||_1=||f_ig_i-f_ig+f_ig-fg||_1\leq||f_ig_i-f_ig||_1+||f_ig-fg||_1 = ||f_i(g_i-g)||_1+||g(f_i-f)||_1\leq||f_i||_p||g_i-g||_q+||g||_q||f_i-f||_p$   (by Holder's inequality) which goes to $0$ as $i\rightarrow\infty$.  However the second part of the question throws me off because I don't see why this proof doesn't work just as well for $p=1,q=\infty$.  Am I missing something? Does the second part also hold? I've had  on the web since 1999, and I'd like to add a page on why complex numbers are (or should be) considered to be numbers.  I'm frequently asked that question.  This is not a question about the existence or usefulness of complex numbers as they do exist as mathematical entities and they are very useful.  It is a question about what makes them numbers.  It could be broadened to ask what constitutes numbers, but I'm specifically interested in complex numbers.  Perhaps it's just historical and sociological, but I'm hoping for an answer that there's something intrinsic about complex numbers that makes them fit into the category of number. Microdisplacements with adaptive subdivision turning black I need to convert one into 1, two into 2 and so on.  Is there a way to do this with a library or a class or anything? What did Hisoka do in the Hunter Exam? In the first Hunter Exam of the series (Hisoka's second),  was introduced into the series removing another applicant's arms from his body for bumping into him and not apologizing.    What did he actually do? We know Hisoka's abilities are Bungee Gum and Texture Surprise, but they don't explain how the arms are converting into flower petals as seen above. What is the capacity of this circuit? I simply can't figure out what the capacity of this circuit is.  I can do the math myself, I just need a hint how to create an equivalent circuit where it is obvious what is parallel and what is in series...