Estimate the lower bound of the following integral.
If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$
Strategy for "Which is better" questions In questions that are asking:    "Which is better option A or option B?"  It seems that several close reasons immediately come to mind:   Primarily Opinion Based.  In many cases the choice between A and B is simply a personal preference, a "6 vs. half dozen", either approach will work just as well and answers to this type of question will be almost entirely opinion based. Too Broad.  In some cases A is better and in others B is better, to go through every possible use case would lead to too many possible answers or one answer that reads like a short text book. Off-topic, Minimal Understanding, include attempted solutions.  This is perhaps the the most important one. If the OP needs to know "Which is better", they should probably have tried both and decided which is better, based on a practical trial.   Yet another option would be to downvote and/or comment with a "Try it and find out".  So...  Are there cases when these questions are permissible?   Could we add some more specific guidance in the help center?  Or should we continue to handle these case by case and just downvote/close appropriately when they pop up?  Strongly related, but hopefully not a duplicate of:   "Which is faster" isn't the only case where these pop up. When is it appropriate to use that as opposed to which with relative clauses? What should one look for when buying a used film camera? I'm planning to buy a used Nikon F5.   I have no experience with a film camera and I want to know what all should one be careful of before buying a used film camera and more specifically a Nikon F5. Why don't cameras show an "accurate" histogram? Why don't cameras show a histogram based on the RAW data rather than on the JPG preview?  I know that a RAW file is not a viewable image and can't be displayed without applying curves and defining how the image should be interpreted. But why don't cameras show a histogram that accurately shows how much latitude I have in the image?  For example when shooting high contrast scenes (for example at sunrise) I know that I can overexpose the highlights to a certain extent to retain detail in the shadows and bring those areas that appear clipped in the JPG preview back in post.  But every camera I've worked with shows me the histogram based on the JPG preview and the picture style and tells me that those areas are clipped.   Is there no way of seeing the actual dynamic range captured by the sensor? Why does no camera manufacturer implement that? It shouldn't be too difficult. Is there a way to "auto login" in PuTTY with a password? Is there a way to configure a password for a stored session in ?  I know there is the capability to specify an "auto-login username" (under Connection/Data), but is there a way to do the same with the password? Do fluorescent lighting and shutter speed create a problem with color cast? I took a bunch of photos at my son's martial arts club using a D90 with a 50mm F1.8.  The club is lit with overhead fluorescent lighting, and I was getting some weird results where some shots are "white", others have an "orange cast", and still others are partly white and partly orange in the same shot.  I suspect it has to do with the combination of the frequency of the light flicker and with the camera shutter speed. Here is a sequence of 3 shots that show the problem.  The shots were taken with 1/500s at F2 and ISO-800 and with Auto-WB.  First shot ("white/normal"):    Second shot ("orange cast" at top):   Third shot ("orange cast" at bottom):   These were taken in burst mode within a second of each other.  Can anybody tell me what is going on?  And, how I can avoid this? I have some simple python code that searches files for a string e.g. path=c:\path, where the c:\path part may vary. The current code is:  def find_path(i_file):     lines = open(i_file).readlines()     for line in lines:         if line.startswith("Path="):             return # what to do here in order to get line content after "Path=" ?   What is a simple way to get the text after Path=? I'm trying to typeset a table that would have a header in the first row (I'm not looking for a different/better solution to adding a header, but for a fix to this solution. Thanks). I want to achieve this by having a multicolumn cell spanning all the columns of the first row. The problem is that if the text is too long, the column widths are uneven, with the last column accumulating excess width. I would expect all of the columns to be of approximately the same width (although I cannot find a decent source that would say they should), given the fact that all the text in the second row is of approximately the same width.  I find this behaviour quite puzzling and have no idea what the cause could be. I know I could perhaps remedy this by manually setting the column widths, but am curious as to how to fix this without doing that (I do not need or indeed want the columns to be of the same width, but roughly proportional to the length of their content), preferably but not necessarily by using the tabular environment only (some other conflict with the document class I am using).   What is it that I'm doing or assuming wrong? And why?  MWE:  \documentclass{article} \begin{document}  \begin{tabular}{ccccc} \multicolumn{5}{c}{This is a piece of text. The longer it is, the longer the last column.} \\  1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\ \hline \end{tabular}  \end{document}   Resulting table. Notice how the last column of the second row accumulates excess width. How to configure Ubuntu as an LDAP client? How to configure Ubuntu 12.04 as LDAP client with PAM modules? prove that if $E$ is connected and $E \subseteq F \subseteq \overline{E}$, then $F$ is connected. Define a set $A$ to be disconnected iff there exist nonempty relatively open sets $U$ and $W$ in $A$ with $U\cap W = \emptyset$ and $A = U\cup W.$ Define a set $A$ to be connected iff it is not disconnected.(there are many equivalent definitions, but I want to prove this lemma using this one). Prove that if $E$ is connected and $E\subseteq F \subseteq \overline{E},$ then $F$ is connected.  Let $U, W$ be a separation for $F$. Find open sets $O_U$ and $O_W$ so that $U = F \cap O_U$ and $W = F\cap O_W.$ I claim that $E\cap O_U, E\cap O_W$ separate $E$. However, I'm unable to show that $U' = E\cap O_U, W' =E\cap O_W \neq \emptyset$ (I think this should be straightforward, but for some reason, I can't figure this out). Suppose $U' = \emptyset.$ Then $E\cap O_U = \emptyset.$ Since $E \subseteq F = U\cup W = F\cap (O_U \cup O_W)\subseteq O_U\cup O_W,$ we have that $E \subseteq O_W,$ so $E\cap O_W = E\subseteq F\cap O_W = W\subseteq F\subseteq \overline{E}.$ Observe that since $E\cap O_U = \emptyset, F\cap O_U = (F\backslash E)\cap O_U\subseteq F\backslash E.$ Similarly, $W'\neq \emptyset.$ Clearly, $U', W'$ are relatively open in $E$. Suppose $U'\cap W' \neq \emptyset.$ Let $x\in U'\cap W'.$ Then $x\in E\cap O_U\cap O_W \subseteq F\cap O_U\cap O_W = U\cap W = \emptyset,$ a contradiction. So $U'\cap W' = \emptyset.$ Also, $U'\cup W' = (E\cap O_U)\cup (E\cap O_W) = E\cap (O_U \cup O_W)$ and $E\subseteq (O_U \cup O_W),$ so $U'\cup W' = E.$ I'm trying to install pandas. When I do, it's automatically installing to the python installation for v 2.7.   How do I force pandas to install to the python 3.5 installation? If $p$ and $q$ are distinct primes , prove that $$p^{q-1}+q^{p-1} \equiv 1 \pmod {pq}$$     Using Fermat's Theorem we can get $$p^{q-1} \equiv 1 \pmod q, \qquad q^{p-1} \equiv 1 \pmod p.$$  After that I have no idea at all. Can anyone help me please How do I install LibreOffice 4? I want to download and install .  How can I do this? How to open and convert CHM documents? I have some documents that are in a .chm format. I wondered if there is a file format that can be easier to navigate, supported and of equal file size in Ubuntu?  If there is, I would like to start converting all those books and probably using them with less hassle on all my Ubuntu PCs and my Android phone. Some observations I made is for $\frac{a}{b}+\frac{b}{a}$, is either:   the denominator has to be one, the numerator has to be a multiple of the denominator or  the numerator and denominator have to be the same.    Obviously, with the given conditions case 3 is eliminated since  $a \neq b$.  For case 1, if $b=1$, then $a=1$ which is a contradiction to the given condition that says $a \neq b$.  For case 2, I would think of examples. For example if $b=2$ then a multiple is $4$, so $a=4$. Then we have:  $$\frac{a}{b}+\frac{b}{a}=\frac{4}{2}+\frac{2}{4}=\frac{10}{4}$$  which is not an integer, but how would i proceed to show this case for all integers. What is the difference between ethics and morals? I am currently writing an essay on artificial moral agents, and I need to explain the difference between ethics and morals. However, I am finding it really hard to find a good book or research paper which contains a citable definition.   Could someone point me in the right direction? Calculate Euler equations of fluid dynamics without division? I'm working on the calculation of the  with the . Unfortunately I'm not allowed to do a division. So I'm wondering if there's a form which does not need a division.  At the moment the Euler equations look like this:  $$ \frac{\partial}{\partial t} \begin{pmatrix} \rho \\ \rho v_1 \\ \rho v_2 \\ \rho v_3 \\ \rho E \end{pmatrix} = -\mathrm{div} \begin{pmatrix} \rho v_1 &amp; \rho v_2 &amp; \rho v_3 \\  \rho v_1^2 + p &amp; \rho v_1 v_2 &amp; \rho v_1 v_3 \\  \rho v_2 v_1 &amp; \rho v_2^2 + p &amp; \rho v_2 v_3 \\  \rho v_3 v_1 &amp; \rho v_3 v_2 &amp; \rho v_3^2 + p \\  (\rho E + p) v_1 &amp; (\rho E + p) v_2 &amp; (\rho E + p) v_3 \end{pmatrix} $$  As you can see, I first need to calculate $\frac{\rho v_1}{\rho}$ to get $v_1$ so I can calculate e.g. $\rho v_1^2$ Is it bad behavior to add filler words such as "so", "um" in business speak? PrintWriter and PrintStream never throw IOExceptions $x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?