I was given the task show that $f(x):= x^2e^{\sin(x)}$ defined on  $[-\frac{\pi}{2},\frac{\pi}{2}]$ has three extrema.   I also have to determine of which kind they are (minimum/ maximum).   I plotted the graph of $f$ and this confused me since according to the graph, $f$ has just $2$ extrema for $x \in[-\frac{\pi}{2},\frac{\pi}{2}]$.
I'm having a hard time finding the solution of the following equation:  $$0 = x\cos(x)+2$$  This is part of showing that $f(x):= x^2e^{\sin(x)}$ has 3 extremas in $[-\frac{\pi}{2},\frac{\pi}{2}]$.
Does assumption of normality of each mixture components implies that each margins is normal I just would like to understand some information about the joint normality and the margins. I read that the normal joint distribution almost always implies that the univariate margins are all normal.   My question is:  1- Is this apply for Gaussian mixture models? For example, in the Gaussian mixture model, each mixture component follows a Gaussian distribution. Hence, does it imply that each margin will be a Gaussian distribution as well? "I understand that mixture components are used to describe the dependency structures, and they are the joint distribution. is that correct?  2- I read this from one answer to a question in this site . "So yes, the assumption of joint normality is a sufficient condition for all marginal distributions to be normal, irrespective of the dependence structure. Hence, the theory of Copulas does not affect this result in any way."  However, I really do not understand this: irrespective of the dependence structure. As I understand if the joint distribution in normal then the dependence structure is normal.   Could someone help me with my questions, please? What are the basic rules and idioms for operator overloading? Is there a limit to a total number of months a holder of UK multiple entry visitor visa can spend in UK per calendar year? Is there a way to set transparency of windows in GNOME? Sometimes ago I used KDE 3.5 I think, and really liked the transparency option for windows, which gave all my text editor a really killer look and feel, really nice/ethereal environment for coding.  I haven't managed to find such a possibility with GNOME.  Any idea if it is possible and how? We have the following points: $$ (0,0)(1,51.8)(1.9,101.3)(2.8,148.4)(3.7,201.5)(4.7,251.1)  \\ (5.6,302.3)(6.6,350.9)(7.5,397.1)(8.5,452.5)(9.3,496.3) $$ How can we find the best fitting line $y=ax$ through the points? My calculator has the option to find the best fitting line $y=ax+b$ through these points, which is:  $$y = 53.28x + 0.37$$  How can I find the best fitting $y=ax$? It seems to me we can't just remove the $0.37$ without compensating in the $a$? Show that $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$ is linear and continuous Let $k:[0,1]\times[0,1]\rightarrow \mathbb{R}$ continuous and $K:C[0,1]\rightarrow C[0,1]$, given by $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$. Prove that $K$ is continuous. I try to see continuity in $0$, but i can't find $\delta$. Let $R$ be a commutative ring with identity. Assume that $R$ has exactly three distinct ideals: $\{0\},I, R.$      1) Show that if $a \in R-I$, then $a$ is a unit in $R$.      2) Let $a,b\ne0$ in $I$. Show that $ab=0$. We walk or ride on our vehicles to our destinations daily. Does our movement have any effect on the rotation of the earth according to Newton's law? What will be the effect if we move all the peoples along with their vehicles at their maximum velocity in one line in one direction along the equator? How much effect will it make? Cant find the solution when sorting an array in JavaScript In general, what is meant by saying that the fraction $x$ of the variance in an analysis like PCA is explained by the first principal component? Can someone explain this intuitively but also give a precise mathematical definition of what "variance explained" means in terms of principal component analysis (PCA)?  For simple linear regression, the r-squared of best fit line is always described as the proportion of the variance explained, but I am not sure what to make of that either. Is proportion of variance here just the extend of deviation of points from the best fit line? Why didn't the Order of the Phoenix assist with the trio (i.e.  Harry Potter, Ron Weasley and Hermione Granger) in the Horcrux hunt? I need to match a space character in a PHP regular expression. Anyone got any ideas?  I mean like "gavin schulz", the space in between the two words. I am using a regular expression to make sure that I only allow letters, number and a space. But I'm not sure how to find the space. This is what I have right now:  $newtag = preg_replace("/[^a-zA-Z0-9s|]/", "", $tag); How do i create a linestring from points sort by date in PostgreSQL?  I tried:  SELECT observations.id,    ST_MakeLine(observations.geom ORDER BY observations.date) AS newgeom    FROM   observations     GROUP BY    observations.id;   This is going wrong. It returns me a linestring with same coordinates.  So, instead of a line from a to b, it gives me a to a.    What am I doing wrong? How can I install more recent versions of software than what Debian provides? Suppose I want a more recent version of software than is available for my current version of an operating system, what can I do?  Cases to consider:   There are semiofficial/official sources of additional packages available for that version of the OS. E.g. backports.org for Debian or PPAs for Ubuntu. There are no more recent versions of the package available for that version of the OS, but there are more recent versions available for more recent versions of the OS. This is the standard case for backporting. There are no packaged versions of more recent versions of the software available. Options available are to package the more recent version.     Per  this is intended as a place to put canonical answers for the following. Answers should probably be made community wiki. I am using an external application to enter data into a Google Spreadsheet.  This workbook is then referenced by a separate workbook (with the IMPORTRANGE(spreadsheet_key, range_string) function) that manipulates and interprets the data.  My problem is that whenever I personally modify the referenced workbook, the referencing workbook updates just fine, but whenever the external application modifies it, it does not get updated.  I have tried editing the "Spreadsheet settings" so that recalculation is done every minute and on every change.  Furthermore, I have also installed an extension in Google Chrome that auto-refreshes the page every hour.  However, the data does not get re-imported from the referenced workbook.  Even if I copy the formula into a new cell, the data is still not re-imported.  Is there anything that I can do to the referencing workbook to trigger Google Sheets to re-import the data?  Edit: Just to be clear, I currently have one workbook with data entered by an external application in it (call it the "source sheet"), and another workbook with an IMPORTRANGE function in it (call it the "referencing sheet").  The data shown by the IMPORTRANGE function in the "referencing sheet" does not include any data entered by the external application since I last personally edited the "source sheet".  Also, both workbooks are using the new Google Sheets.  Edit: Also, this question is not the same as  because I am using the function given as the solution to that question to import data from a spreadsheet.  The problem is not how to import the data, but rather how to update the source for the data.  My assumption would be that Google would have taken care of this for me, but the data in the "referencing sheet" does not get updated, and the only way I can find to get it to update is to physically go into the "source sheet" and edit it myself. How to break out of jQuery each Loop Why CAN we see the new moon at night? I understand that the Moon's phases are determined by its position in orbit relative to the Sun. (See: ). The "shadow" is not cast by the Earth (a common misconception - this is actually a lunar eclipse), but by the moon's body itself.  It would appear that, in order for the moon to be a new moon, it would have to be somewhere between the Earth and the Sun. However, it would seem that, if we were looking at a new moon, we'd necessarily be looking at the Sun as well.  Why, then, can we sometimes see a new moon at night? Why doesn't it vanish at night for a half-month every month, between its last/first quarters? I am using VC++. Is assert(false) ignored in release mode? cardinality of the set of $ \varphi: \mathbb N \to \mathbb N$ such that $\varphi$ is an increasing sequence I know that the set of functions $ f:\mathbb N \to \mathbb N$ is uncountable, but what if we consider only $f$ such that $f$ is increasing? I want to know if this set is countable D: and also the case of bijective and increasing (clearly if the firstone holds then this too). I think that it could be possible that it's countable Move iTunes folder outside Music folder iTunes by default stores its settings, the iTunes Library.itl file, and the music library audio tracks, inside the \Users\username\Music\iTunes folder.  I don't want my Music folder polluted with program settings, especially because I'm synchronizing that folder on the cloud. Via the settings panel I can only change the path of the audio tracks, but the settings remain in the default position.  Is there a way to completely move the iTunes folder to another path?  PS: I've looked at the answers to , but the focus there is on moving the media library only, not the whole iTunes directory.