Let f be a continuous strictly increasing function from $[0,\infty) \rightarrow [0,\infty)$ and $g=f^-1$ (i.e.$f(x)=y \iff g(y)=x$). Let $a,b&gt;0$ and $a\neq b$.Then $ \int\limits_{0}^{a}f(x)dx$+$\int\limits_{0}^{b}g(y)dy$ is (A) greater than or equal to ab (B) less than ab (C) always equal to ab
Let $f:[0,\infty)\rightarrow [0,\infty) $ be a continuous strictly increasing function and $g=f^{-1}$. Let $a,b&gt;0$ and $a\neq b$. Then how $\displaystyle\int_{0}^af(x)dx+\int_0^bg(y)dy\geq ab.$ Any hint will be appreciated.
Use of each and every Why is there conservation of kinetic energy in elastic collision and not in inelastic collision? How to calculate polygon areas and perimeters using QGIS? Using QGIS, if I load a polygon layer and make a new column, how do I fill that column with the:   area of each polygon and another column with length of the perimeter of each polygon? Would a monk with the Tavern Brawler feat have improvised weapons count as monk weapons?  It states in the monk's Martial Arts feature (PHB, p. 78) that:     At 1st level, your practice of martial arts gives you mastery of combat styles that use unarmed strikes and monk weapons, which are short swords and any simple melee weapons that don’t have the two-handed or heavy property.   The Tavern Brawler feat (PHB, p. 170) states that:     You are proficient with improvised weapons and unarmed strikes.   And the rules for improvised weapons (PHB, p. 147-148) state:     In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus. An object that bears no resemblance to a weapon deals 1d4 damage.    If the improvised weapon does not resemble a weapon, is simple, and is not heavy or two-handed, could a monk replace the d4 dice? For example if the monk is level 6 and has a unarmed attack of a d6, with a d6 dice instead? I've recently updated to Ubuntu 15.04 and after installing the latest nvidia drivers (346.59) which worked on ubuntu 14.10 seems to have caused some trouble on boot.  All I get is the following when booting:   A purple screen for about a second and then  [0.514409] ACPI PPC Probe failed. Starting version 219  A black screen with a message of my monitor saying : No Signal detected.   I am using a GTX 970. Ruby Style: How to check whether a nested hash element exists I'm self-studying a bit of complex analysis, and I'm attempting to figure out the following.   Suppose $\lim_{n\to\infty}z_n=A$. How can I show that $$ \lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A. $$  Is there a clever way to write the limit to make it more approachable? Thank you. Gedit has a transparent background since upgrade to 16.10 Since I upgraded to from 16.04 to 16.10, gedit has strange new "feature": its background is totally transparent.    Where can this come from ?  Notice I'm using Unity and don't have any specific gnome package installed besides what comes by default. Notice also that in 16.04 I used the Vivacious theme from ravefinity PPA, but I had to go back to Ambiance as Vivacious was broken in 16.10. Maybe it's related.  [EDIT] changing the 'Fonts and Colors' in gedit to any theme other than default (Kate, Solarized, Oblivion, etc..) solves the issue. So there must be a faulty gtksourview file in my home somewhare, but I can't find it... Prove that $\lim f(x) =0$ and $\lim (f(2x)-f(x))/x =0$ imply $\lim f(x)/x =0$ When using the serial console of my system I always end up with $COLUMNS=80 and $LINES=24.  While I can change these variables manually it is somewhat annoying to do this any time when the client side terminal window has been resized.  Usually I'm connecting to the console using screen /dev/mytty baudrate.  Changing the $TERM environment variable to "screen" or "xterm" does not help.  Will I need to call getty with some of those instead of vt100?  Needless to say that all of this works fine, when I connect to the same machine using ssh. Force refresh view in LWC In Aura, we can do      $A.get('e.force:refreshView').fire();   to cause standard components to update.  How can we do an equivalent in LWC because sfdx refuses to publish .js code with $A in it saying the usage is disallowed in LWC?  For now i tricked it with   eval("$A.get('e.force:refreshView').fire();");   but that is obviously less than ideal.... Heterogeneous vs. inhomogeneous Has it been practically proven that quantum superposition exists ? If yes, how does it even work? I would like to customize Ubuntu live CD by installing some additional packages. I have followed   but it doesn't seems to work. Can anyone provide clear instructions?   Note: I do not prefer Remastersys, manual way will be appreciated.  Customization  Packages that I want to install:   Thunderbird    Samba    SSH       Changes that I need:   Remove Games menu from the Application menu Firefox shortcut on Desktop Radiance as the default Theme Different default Ubuntu Wallpaper   Configuration file changes   I want the panel to be placed at the bottom I want to paste my Samba configuration file instead of default Samba configuration I have few Firefox shortcuts and folders I would like to show that in Desktop Also it will be nice if you say me how to change the icon sets   Recent Updates   I have customized Ubuntu 10.10 with Firefox shortcuts and few folders on desktops. Everything went smooth. But the installer gets crashes after choosing the timezone. How do i fix this issue? Also setting wallpaper affects the login screen. The wallpaper which i set is displayed on the login screen also. I just want the default one for the login screen. How to calculate: $\sum_{n=1}^{\infty} n a^n$ I've tried to calculate this sum:  $$\sum_{n=1}^{\infty} n a^n$$  The point of this is to try to work out the "mean" term in an exponentially decaying average.  I've done the following:  $$\text{let }x = \sum_{n=1}^{\infty} n a^n$$ $$x = a + a \sum_{n=1}^{\infty} (n+1) a^n$$ $$x = a + a (\sum_{n=1}^{\infty} n a^n + \sum_{n=1}^{\infty} a^n)$$ $$x = a + a (x + \sum_{n=1}^{\infty} a^n)$$ $$x = a + ax + a\sum_{n=1}^{\infty} a^n$$ $$(1-a)x = a + a\sum_{n=1}^{\infty} a^n$$  Lets try to work out the $\sum_{n=1}^{\infty} a^n$ part:  $$let y = \sum_{n=1}^{\infty} a^n$$ $$y = a + a \sum_{n=1}^{\infty} a^n$$ $$y = a + ay$$ $$y - ay = a$$ $$y(1-a) = a$$ $$y = a/(1-a)$$  Substitute y back in:  $$(1-a)x = a + a*(a/(1-a))$$ $$(1-a)^2 x = a(1-a) + a^2$$ $$(1-a)^2 x = a - a^2 + a^2$$ $$(1-a)^2 x = a$$ $$x = a/(1-a)^2$$  Is this right, and if so is there a shorter way?  Edit:  To actually calculate the "mean" term of a exponential moving average we need to keep in mind that terms are weighted at the level of $(1-a)$. i.e. for $a=1$ there is no decay, for $a=0$ only the most recent term counts.  So the above result we need to multiply by $(1-a)$ to get the result:  Exponential moving average "mean term" = $a/(1-a)$  This gives the results, for $a=0$, the mean term is the "0th term" (none other are used) whereas for $a=0.5$ the mean term is the "1st term" (i.e. after the current term). Is the following layout possible with the subfigure package? Correct usage for "all you need is/are..." Enable Wireless Monitor mode for traffic sniffing I need to sniff all the wifi traffic that I can capture using wireshark in Ubuntu. My wireless interface is named as eth1 in the interfaces list. When I select that interface in Wireshark and start new sniff, I got following error;    As  thread says, I tried to installed the mentioned driver but I got an error in installation saying that that driver is not supported by my device. After that I tried the second answer in the same thread and run following command to enable monitor mode in my wireless card.  echo 1 &gt; /proc/brcm_monitor0   Then I saw a new Ethernet interface (not a wireless interface ) called prism0 in wireshark interface list. I couldn't start a sniff using that interface using monitor mode because in that interface settings, monitor mode check box has been disabled.  I have installed a broadcom wireless driver in Additional Drivers as follows:    Is there an error listing my Wifi interface as eth1 rather than wlan0 in the interfaces list? How can I start my wireless interface in monitor mode for sniffing?   Thank you. Is verbatim copying several paragraphs of text with citation considered plagiarism? Is it plagiarism if I copy several paragraphs from another's source (let's say a CS paper) into my work and then footnote Source: ..., for example, as background information? Non-existence of $\lim \limits_{(x, y) \to (0,0)} \frac{x^3 + y^3}{x - y} $ How to show that $\lim \limits_{(x, y) \to (0,0)} f(x, y)$ does not   exist where,         $$f(x, y) =  \begin{cases} \dfrac{x^3 + y^3}{x - y} \; ; &amp; x \neq y \\  0 \; \;\;\;\;\;\;\;\;\;\;\; ; &amp; x = y  \end{cases}  $$   I tried bounding the value of the function as $(x, y)$ approaches $(0,0)$ but was not successful. I graphed the function and saw that it was actually approaching $0$ but Microsoft Math failed to render some points around the $x = y$ plane. So I imagine I need to find points $(x, y)$ such that $x$ is extremely close to $y$ but $(x, y)$ is not as close to $(0,0)$ to show that the function does not approach $0$ around the origin. But was unable to think of any.   Hints or suggestions would be awesome. Any help is appreciated.