What is wrong with this solution of find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$ What is wrong with this solution of find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$  They all are positive terms so arithmetic mean is greater than equal to geometric mean. $$ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x\geq  3(  \sec^6 x \csc^6 x \sec^6 x\csc^6 x)^\frac{1}{3}   $$  $$ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x \geq 3(  \sec x \csc)^4 $$  $$ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x\geq  \frac{3 * 2^4}{\sin ^4 2x} $$ Clearly least value is 48, but something is wrong here, as the answer is 80, if I use other methods.
Find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$ Find the least value of $$ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$$ I tried AM greater than equal to GM But that's for finding maximum value.  This can probably be solved with calculus but I don't know for some reason I can't find the answer . Which is $80$.
Is it possible to know who voted my question/answer up or down? If not, then is it possible to include this feature? By this feature we can come to know if you are being voted up by question owner or persons giving other answers to same question or just the visitors of that question thread.  Like...  If it is a group of people giving other answers, then your answer is most correct If it is readers of that question, then your answer is more popular Given this (very) tricky determinant, how can we calculate it easily?  $$\begin{pmatrix} \alpha + \beta &amp; \alpha \beta &amp; 0 &amp; ... &amp; ... &amp; 0 \\ 1 &amp; \alpha + \beta &amp;  \alpha \beta &amp; 0 &amp; ... &amp; 0 \\ 0 &amp; 1 &amp; \alpha + \beta &amp; \alpha \beta &amp; ... &amp; ... \\ ... &amp; ... &amp; ... &amp; ... &amp; ... &amp; 0 \\ ... &amp; ... &amp; .... &amp; ... &amp; ... &amp; \alpha \beta \\ 0 &amp; 0 &amp; 0 &amp; ... &amp; 1 &amp; \alpha + \beta   \\ \end{pmatrix} \in M_{n\times n}$$  EDIT:     I have to prove it is equal to $\frac{{\alpha}^{n+1} - {\beta}^{n+1}}{\alpha - \beta}$   Any help is appreciated, I just could not find a trick to ease it up! Any situation, any time; this person has always something negative to say about it.  It should be a noun or an adjective to describe a person who is a damp cloud walking over every sunshine situations at the cubicle, hoping to drench other people with his miserable comments.    I considered "sourpuss" but would prefer something more important sounding, like "misogynist." When (and why) should you take the log of a distribution (of numbers)? I recently started an account on , and went to add an answer to a question which was protected. I have enough rep on other sites that I have the "Association Bonus", so I was confused as to why I wasn't being allowed to answer the question.     This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation on this site.   After re-reading the message and reviewing , I realized that the bonus didn't count as points earned toward that site.  This is a really minor issue, but I think it's worth fixing (not sure if bug or feature).  I believe that users who have enough rep to receive the association bonus should generally already know not to post "thanks!" and "me too!" style answers, so if the message is accurate, users who receive the association bonus should be able to answer protected questions.  If there's another reason to keep users from answering protected questions, such as  claim that:     For protected questions, you are expected to have experience on the target site itself. Experience at other sites is not enough.   then the message should be updated to reflect this.  Option 1:  make protected questions open to anyone who has >10 rep on that site and update the message to read:     This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have at least 10 reputation on this site.   Option 2:  leave protected questions the way they are, but update the message to read:     This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation on this site to show that you have experience on this topic; experience on other StackExchange sites is not enough.   Option 3:  Answer with your own suggestions/improvements to these ideas. Can any one help? I purchased a solar powered PIR security light, with the intention of putting it inside my kitchen window. The PIR would be inside the window, but this does not work.  When I go outside to activate the light, it does not work. When I go inside to run my hand across the sensor, it works.  Will a PIR unit penetrate glass, or am I barking up the wrong tree? When is "self" required? I have been using classes for only a short while and when I write a method, I make all variables reference self, e.g. self.foo.  However, I'm looking through the wxPython in Action book and notice that "self" isn't used all the time. For example:   import wx  class TextFrame(wx.Frame):     def __init__(self):         wx.Frame.__init__(self, None, -1, 'Text Entry Example',             size=(300, 100))         panel = wx.Panel(self, -1)         basicLabel = wx.StaticText(panel, -1, "Basic Control:")         basicText = wx.TextCtrl(panel, -1, "I've entered some text!",             size=(175, -1))         basicText.SetInsertionPoint(0)         pwdLabel = wx.StaticText(panel, -1, "Password:")         pwdText = wx.TextCtrl(panel, -1, "password", size=(175, -1),             style=wx.TE_PASSWORD)         sizer = wx.FlexGridSizer(cols=2, hgap=6, vgap=6)         sizer.AddMany([basicLabel, basicText, pwdLabel, pwdText])         panel.SetSizer(sizer)   The one below does use "self".  import wx class ButtonFrame(wx.Frame):     def __init__(self):         wx.Frame.__init__(self, None, -1, 'Button Example',             size=(300, 100))         panel = wx.Panel(self, -1)         self.button = wx.Button(panel, -1, "Hello", pos=(50, 20))         self.Bind(wx.EVT_BUTTON, self.OnClick, self.button)         self.button.SetDefault()     def OnClick(self, event):         self.button.SetLabel("Clicked")   If I remember correctly, "self" is reference to a particular instance of the class, so when is it not necessary? Is there a general rule of thumb? How can I minimise the time to get a new UK passport? How to remap an existing keybinding in Zathura The USB has various purposes for data transfer, movement of input and output signals for hardware like connecting an Android device to another Display screen (USB on the go), charging the device etc.(during a device charge the voltage drawn from the USB is same irrespective of the device or OEM then Why are two USB Cables different ?  A USB of one manufacturer may not be supporting another manufacturer. We get problems if we do connect them like      or Here is my attempt to prove the Well-Ordering Principle, i.e. that any non-empty subset of $\Bbb N$, the set of natural numbers, has a minimum element.  Proof. Suppose there exists a non-empty subset $S$ of $\Bbb N$ such that $S$ has NO minimum element. Define $A = \left\{n\in \Bbb N : (\forall s\in S)(n \leq s)\right\}$. It is obvious that $1\in A$. Suppose $n\in A$, then for each $s \in S$, there exists $q \in N$ such that $n + q = s$. Since $q \ge 1$, $n+1 \leq s$, for all $s\in S$. By Principle of mathematical induction, $A = \Bbb N$. Take any $s_0 ∈ S$, then $(\forall s\in S)(s_0 \leq s)$. (This contradicts that $S$ has no minimum element).  How do I prove the statement without invoking Mathematical Induction? Also, I read that the proof of Principle of Mathematical Induction makes use of Well-Ordering. Can it be proven independently of Well-Ordering too? How do you write (and run) a correct micro-benchmark in Java?  I'm looking for some code samples and comments illustrating various things to think about.  Example: Should the benchmark measure time/iteration or iterations/time, and why?  Related: Last night I was grilling chicken thighs on a barbecue and I was told that if there was any pink inside them, they are not safe to eat. My question is twofold:  (a) Is this true?  (b) Is there a way to tell that doesn't involve cutting the thighs open to have a look at the colour? I am using an old laptop and doing heavy processing that needs high CPU usage for a long time (~30mins - 2hours). When the process puts heavy load on CPU for a long time, CPU overheats and computer shuts down. I don't want to use a cooler/stand because I am carrying this laptop around and working outside home.  Is there a way/application that I can limit CPU usage of a process in Windows?  Note: Setting the priority of the process from task manager does not work.  Note 2: Fans are clean. Help for Mathematics High school I'm looking for mathematics books, courses + exercises for high school level. My daughter was high school certificated but she has stopped her studies for 5 years. She would like to be involved in new educational program but she needs to restore her mathematic capabilities.  Could you help me to identify which mathematics books can help her?  Thanks a lot for your reply,  FR. NO2 CSV downloaded files shows zero values for consider date range using Google Earth Engine When I am downloading the CSV file, the results show zeros in the CSV file. How can I download exact NO2 values as shown in Chart after running the mentioned below code in Google Earth Engine? // Create a FeatureCollection from the list and print it. var BergamoGeometry = ee.FeatureCollection(BergamoGeometry); print(BergamoGeometry); Map.centerObject(BergamoGeometry); Map.addLayer(BergamoGeometry); //////////////////////////////////////  var dataset = ee.ImageCollection(&quot;COPERNICUS/S5P/NRTI/L3_NO2&quot;)                  .filterBounds(BergamoGeometry)                   .filter(ee.Filter.date('2019-03-30', '2019-04-29'));                   var NO2 = dataset.select('NO2_column_number_density'); var band_viz = {   min: 0,   max: 0.0002,   palette: ['black', 'blue', 'purple', 'cyan', 'green', 'yellow', 'red'] }; // Create and print the chart. print(ui.Chart.image.series(NO2, BergamoGeometry, ee.Reducer.mean(), 30)); I'd like to do some URL rewriting (Why? See .) so that instead of users seeing addresses like   labouseur.com/course-compilers.html    they can instead see and use simply   labouseur.com/course-compilers   (Even better, maybe I should restructure that so that it's courses/compilers.)  I'm using a Linux-based shared hosting service for my website, so I do not have administrative control of the server, but I do have control over .htaccess. The references I've read online seemed less than clear to me, so I'm looking for a little clarity and advice here. ASP.NET MVC 3 localization with DisplayAttribute and custom resource provider Is your MAC address revealed when you hit up a website? When you hit up a site, such as cnn.com, does the TCP/IP packet contain the MAC address of your network card?  Could cnn.com theoretically record my MAC address? Is openssl rand command cryptographically secure? I'm wondering if the openssl rand command produces cryptographically secure random bytes. For example when in need for a random password or token:  openssl rand -hex 32   The man page unfortunately does neither state it's cryptographically secure, nor that it's not.  On the one hand, I think this is openssl, its sole purpose is to do cryptography. On the other hand, the man page states something about reading and writing seed data instead of directly querying a CSPRNG.  So it is somewhat unclear for the end user. Does someone know?