How to prove that the equation $x^5-1102x^4-2015x=0$ has at least three real roots? I've already used the Intermediate Value Theorem to show that there is at least one real root. $f(1)=-3116&lt;0$ $f(-1)=912&gt;0$ Using IVT to $N=0,$ there exists $c \in (-1,1)\;$ such that $\;f(c)=0.$ I can use Rolle's Theorem to show that there is at least $2$ real roots, but how do I show that the equation has at least three real roots?
Real Roots and Differentiation Prove that the equation $x^5 − 1102x^4 − 2015x = 0$ has at least three real roots.  So do I sub in values of negative and positive values of $x$ to show that there are at least three real roots? The method to do this question is not by finding the factors of $x$ right? Because it will be too tedious so I want to ask whats the other solution to prove this? Help appreciated. Thank you very much.
Why did Itachi poke Sasuke's forehead? From the beginning of the series, even during his last moments, Itachi poked Sasuke's forehead as a way of showing affection and apologizing.  But is that the only reason? Did he perhaps try to transfer some knowledge or memories to Sasuke? Or is it simply unknown? Why is everything in minecraft opaque red and blue? How to vertically center content with variable height within a div? Why aliases in a non-interactive Bash shell do not work Motor to spin slowly, low torque Screen (not Lock Screen) Timeout Stuck at 1 Minute I recently got a new laptop (Lenovo T440p with Windows 8.1 Pro) and was having the following issue: the screen would lock (screen would go black and require a password to get back in) after 1 minute. I would prefer something more like 10 or 15 minutes. So, I tried the obvious changes (check the screensaver settings, power management settings for sleep, turning off the screen, etc.), with no success. I did, however, notice a rather odd behavior: if I changed the screensaver timeout from 1 minute to anything else (say, 15 mins), clicked OK, and re-opened the screensaver settings window, it would be back to 1 minute. Odd.  At that point, I turned to Google. I found posts ( in particular was helpful) suggesting all sorts of causes and resolutions:   If you have Cisco AnyConnect VPN client (which I did), try uninstalling it. I uninstalled, rebooted, no change. Open the User Accounts control panel and, on the left side of the screen, click Reset Security Policies. My User Accounts control panel, however, has no such link. Reboot into a repair console and run "sfc /scannow". It did indeed find and repair some issues, but didn't fix the screen timeout problem. Edit the Group Policy to force a particular timeout time; the setting showed up correctly (15 mins) in screensaver settings, power management settings, etc., but the screen would still go black and lock after 1 minute. In the registry, manually change the screensaver timeout and the "turn off the screen" timeout. No luck. Simple question:     My apples and orange are wrong   or     My apples and orange is wrong   I am not a native English speaker, and I am having some trouble choosing between plural are or singular is for that kind of example. Literature: 'Why' at the beginning of sentences I’m currently reading George Martin’s A Song Of Ice And Fire novels in English. As a non-native speaker (I’m German), I stumbled upon some grammatical constructs that I’ve never seen before, one of them being the Why at beginning of sentences.  Examples:   Tyrion gave the rotting head a second look. Why, it almost looks as if those lips are smiling. Did he use that word? Why, the boy has a singer's soul . . . though if you believe that song, you may well be dimmer than the first Reek.   These kind of sentences are obviously different from sentences like Why is the sky blue?  I’d like to know if that construct is archaic, or could be used in everyday language, and how one could rephrase that kind of sentences. Can it be rephrased at all? To me it seems like the Why is just used to give the sentence a kind of “cynical” taste. Viewing all my questions from Android app How can I make a program executable from everywhere What should I do if I want to be able to run a given program regardless of my current directory? Should I create a symbolic link to the program in the /bin folder? I don't know much more about SEO, but it's interesting to learn about it. When I tried to search my website with an associated keyword or you can say with the exact name of the domain, I got some suggestions by Google. Here's .  First of all, I don't want to spamming here, I just want prove my question and want to give a hint about what I am talking here. The above link display the Did You Mean, I know the specific reason why Google shows this, but I want to know the Google's guidelines and some guidance by you experts so that if someone search for the particular keyword then Google hide that Did You Mean and display the result. What sort of cheat sheets for a new DM lead to a better flowing 5e game of Dungeons and Dragons? The question is, what elements on a DM screen are essential? I'm new at playing and DMing, altough a lot of experience watching games, just got a group to play with now. Started with the starter set in 5E and the Lost Mine of Phandelver adventure, however I've noticed the game can come to a stop when checking the rules or how to do something, while getting the books and stuff like that. So I thought to make a DM screen with some cheat sheets, of the most usual things new players, and new DM's have doubts on, and how to make the game flow well without long rule-checking breaks. What are the essential things to have on that screen? So far this is what I found  but not sure if it has all I need or so much stuff that'll also make the game stop a lot while cheking it out. I'm hoping someone could check my work. I'm trying to simplify the equation for $x$. $$(\frac{x}{a})^{3.2} + (\frac{y}{b})^{3.2} = 1$$ $$\frac{a}{b} = \frac{174.1}{86}$$ Thus I began by substituting $a$ with: $$a = \frac{174.1}{86}b$$ $$((\frac{86}{174.1})(\frac{x}{b}))^{3.2} + (\frac{y}{b})^{3.2} = 1$$ $$((\frac{86}{174.1})(\frac{x}{b}))^{3.2} = 1 - (\frac{y}{b})^{3.2}$$ $$(\frac{x}{b})^{3.2} = \frac{(1 - (\frac{y}{b})^{3.2})(174.1)^{3.2}}{(86)^{3.2}}$$ $$x^{3.2} = \frac{(1 - (\frac{y}{b})^{3.2})(174.1)^{3.2}(b)^{3.2}}{(86)^{3.2}}$$ $$x^{3.2} = \frac{(1 - (\frac{y}{b})^{3.2})((174.1)(b))^{3.2}}{(86)^{3.2}}$$ $$x^{\frac{16}{5}} = \frac{(1 - (\frac{y}{b})^{3.2})((174.1)(b))^{3.2}}{(86)^{3.2}}$$ $$x = [\frac{(1 - (\frac{y}{b})^{3.2})((174.1)(b))^{3.2}}{(86)^{3.2}}]^{\frac{5}{16}}$$ Is my work correct, or am I completely off the board? How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$? I can show that the following limit exists but I am having difficulties to find it. It is $$\lim_{n\to \infty} \sum_{k=1}^n \frac{k^n}{n^n}$$ Can someone please help me? Why is $1^{\infty}$ considered to be an indeterminate form Disallow downvoting other answers if I've posted my own Possible Duplicates:        As Dan Dyer mentioned in his comment on , there is a conflict of interest when you have posted your own answer and are allowed to downvote other answers.  Despite the vote editing window being drastically reduced, the fundamental problem remains: I can downvote the other answers permanently and still benefit - my answer gets upvoted ONCE and wham, I have just made up the rep loss for my 10 other downvotes. The vote editing window fixes part of the problem, but there is a more complete fix that has the extra benefit of not preventing legitimate vote changes: to disallow downvotes on competing answers when you have posted one yourself.  The temptation to wrongfully downvote competing answers is huge. The rep loss is tiny in comparison. It has to be in order for people to continue to legitimately downvote.  Details:   If I have previously voted and then post my own answers, delete my downvotes. Continue to allow me to remove the upvotes - I think this is fully legitimate. If I delete my answer it's probably best to continue to deny downvotes (perhaps even including the question), but I don't have a strong opinion either way. Cascading Drop down in SharePoint Online Is it proper to use "the" before the name of a government organization? After my suspension period, my reputation was not returned On Stack Overflow, I have a suspension period of 7 days which ends today. But my actual reputation has not been returned to me as one of the moderators said it would. I have a message like this by @Andrew Barber:     Once your suspension period is over, you will be returned to your   correct reputation level, minus the votes from the accounts which have   been deleted. At that time, you are more than welcome to return, and   continue contributing your knowledge, and gaining legitimate   reputation points. Just please keep our rules in mind when you do.      Regards      Andrew Barber, Stack Overflow moderator   Can any moderator here help me or tell me what the rules are of getting back my reputation?  How can I get the attention of a moderator so that they can resolve my problem? Comma use with nonessential appositives Sorry, I do not know the language terminology. Is this a case of nonessential appositive?  Which of the following sentence has a correct comma use? Comma before &quot;should&quot;  Importantly, two frail subgroups of stroke population, long-term residential care and cognitive impairment patients, should not be excluded from the rehabilitation.  No comma before &quot;should&quot;  Importantly, two frail subgroups of stroke population, long-term residential care and cognitive impairment patients should not be excluded from the rehabilitation.