How do I show that a real polynomial function of odd degree has at least one real zero? That is when the function is 0 and cuts the x-line.   This is what I have got so far:  f(x) = x^(2k+1)   Can I then put x to zero and that proves it? However, that doesn't feel like I'm proving it for all real polynomial functions of odd degree.
Im asked to prove that a polynomial of an odd degree has a root. Im going to use IVT to prove this, but I'm wondering if I can use the assumption that an odd degree polynomial has either two cases:  $$ \lim_{x\to\infty}h(x)=+\infty \\ \lim_{x\to-\infty}h(x)=-\infty \\ $$ $$and$$ $$ \lim_{x\to\infty}h(x)=-\infty \\ \lim_{x\to-\infty}h(x)=+\infty \\ $$ Then by this there exists numbers $a,b$ such that $$ h(a)&lt;0&lt;h(b) $$ $$or$$ $$ h(b)&lt;0&lt;h(a) $$ Which I think would prove that odd degreed $h(x)$ has at least a root. Or is there a more convincing way to show that there are numbers in $h(X)$ greater than zero and less than zero?  edit: (sorry about the bad limit notation)
Static nested class in Java, why? Every principal ideal domain satisfies ACCP. I'm absolutely new to Linux. I would like to know how to install Ubuntu alongside the pre-installed Windows 8+ OS.  Should I do it with Wubi, or through the Live USB/DVD?  What steps do I need to take to correctly install Ubuntu? Name lookups in C++ templates $f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$? Let $f:\mathbb R \to \mathbb R$ be a continuous function such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , then is it true that $f(x)=x , \forall x \in \mathbb R$ ? I can see that $f$ is bijective , hence strictly monotone . But I cannot progress further . Please help. Thanks in advance String concatenation: concat() vs "+" operator Are old (>35) faculty candidates discriminated against all over the world? I think that in India research institutes will never take a faculty candidate who is over 35. Is it so all over the world?   I am asking this question from the point of view of if someone starts their PhD. at the age of 27 then its unlikely they will be faculty candidates till they are 40. A friend of mine was trying to explain to me how all infinities are equal. For example, they were saying that there are the same amount of numbers between $0$–$1$ as there are between $0$–$2$.  The way they explained it, you could prove that there are the same amount of numbers by getting a match to each number.  For example, for any number in the range from $0$–$2$, you can find a matching number in the range $0$–$1$ by dividing the number by 2. In every case, the ending number will end up between $0$–$1$. In addition, if you multiplied any number in the range of $0$–$1$ by 2, you will always end up with a number between $0$–$2$ Therefore, there should be the same infinite amount of numbers between  $0$–$1$ as there are between $0$–$2$.  Is this principle accurate? The problem that I had with it was, if in-fact there are the same amount of numbers between 0 and 1 as there are between 0 and 2, why is 2 greater than 1? I just installed Ubuntu 14.04.3 LTS after installing windows 10 in my new VN7-571G and the wireless is not working. It never had: even during the installation it was not recognized. I run a script to gather the data , which is . lspci -knn | grep Net -A2 output 03:00.0 Network controller [0280]: Qualcomm Atheros Device [168c:003e] (rev 20)         Subsystem: Lite-On Communications Inc Device [11ad:0804]  The OS behaves like it has no wifi card. I do not see wifi settings in network settings I'm not a linux power user, so I do not know what to do to solve the issue. What should I do to make it work? How do I make money efficiently? In Oblivion, I had two favorite money-makers:   Dump all my stuff except some basic armor and potions, and raid an oblivion tower. The armor and weapons was worth lots, and selling it in Anvil made a lot of cash fast. Harvest an entire farm's worth of crops, create 'restore fatigue' potions and sell them.   Skyrim seems to spoil this:   Merchants have a max-amount per day/QOD/... instead of per item You can't carry as much (or maybe I have yet to discover the strength/feather spells) Alchemy is a lot harder (most common items go into food recipes now, which don't sell for those prices) On top of that, once a dungeon is cleared, it doesn't respawn - so no 'farming' familiar dungeons?   What are some easy/repeatable money making methods in Skyrim? Which are most efficient effortwise/real-life-time wise? Is there a converter that will convert LaTeX to HTML? I have tried several services but the LaTeX diagrams and some of the maths came out weird.  I actually converted the Latex to PDF without a problem. Then I used  to convert from PDF to HTML. The square root signs did not appear properly and some of the Cartesian coordinate plane diagrams had nothing on them except the grid.   \documentclass[12pt,a4paper]{report} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{pstricks,pst-plot} \usepackage{pst-eucl} \addtolength{\hoffset}{0.2cm} \addtolength{\textwidth}{1.1cm} \addtolength{\textheight}{1.0cm} \addtolength{\voffset}{-0.7cm} \linespread{1.3} \begin{document}  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$     $\hspace{30 mm} $     (6)\\  (4) and (6) tell us that\\  (4)  $-1 \leq x \leq 0$  \\  (6)  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$\\  Therefore $-1 \leq x \leq -\sqrt{\frac{1}{2}}$\\  \vspace{50 mm}  \begin{align*} \psgrid[unit=2cm](0,0)(-2,-2)(2,2) \pnode(-1.8,.87){b}\uput[50](b){b} \pnode(-1.8,1.8){a}\uput[50](a){a} \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,4) \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,-4) \psset{linecolor=blue} \psellipticarc(0,0)(2,2){0}{180} \psset{linecolor=red} \psline[linewidth=1pt,linestyle=dashed,dash=2pt](-2,2)(-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt](-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(4,-4) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(-2,2)(-4,4) \end{align*}  \end{document}   PDF output:    PDF->HTML output: How to automatically generate N "distinct" colors? How to test randomness (case in point - Shuffling) First off, this question is ripped out from  question. I did it because I think this part is bigger than a sub-part of a longer question. If it offends, please pardon me.  Assume that you have a algorithm that generates randomness. Now how do you test it? Or to be more direct - Assume you have an algorithm that shuffles a deck of cards, how do you test that it's a perfectly random algorithm?  To add some theory to the problem - A deck of cards can be shuffled in 52! (52 factorial) different ways. Take a deck of cards, shuffle it by hand and write down the order of all cards. What is the probability that you would have gotten exactly that shuffle? Answer: 1 / 52!.  What is the chance that you, after shuffling, will get A, K, Q, J ... of each suit in a sequence? Answer 1 / 52!  So, just shuffling once and looking at the result will give you absolutely no information about your shuffling algorithms randomness. Twice and you have more information, Three even more...  How would you black box test a shuffling algorithm for randomness? Vote too old to be changed, unless post is edited Anyone seen this one yet? Vote too old to be changed, unless post is edited My vote was only a few hours old. Then I realized the answer was wrong. Updated my question and went to remove the vote. Even after updating question I still have the issue. Small issue, but frustrating.  See also: Anonymous questions and Avatars Some Stack Exchange sites allow to ask an anonymous question by specifying a one-time user name and an email which is "required, but never shown". However the Gravatar attached to the email is shown anyway. Anonymous users do not want their picture to appear - someone can find their identity via google image search or someone that knows them.  Example: open incognito window and : How to prove that exponential grows faster than polynomial? In other words, how to prove:  For all real constants $a$ and $b$ such that $a &gt; 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$  I know the definition of limit but I feel that it's not enough to prove this theorem. How to compute $\prod\limits^{\infty}_{n=2} \frac{n^3-1}{n^3+1}$ How to preserve the same parskip in minipage Zero-dimensional ideals and finite-dimensional algebras I encountered in the literature the term zero-dimensional ideal, however I can not find a relevant definition anywhere in Atiyah-MacDonald or Matsumura. In fact, I encounted the statement:      $I$ is a zero-dimensional ideal of $k[x_1,\dots,x_n]$ $\Leftrightarrow$ $k[x_1,\dots,x_n]/I$ is a finite-dimensional $k$-vector space, where we assume that $k$ is algebraically closed.   Question 1: What is the definition of the dimension of an ideal?  Question 2: How do we prove the above statement?  Question 3: Is $k$ being algebraically closed a necessary condition for the statement to be true? I am using ArcGIS Desktop 10.1.   I am trying to find the location of the highest value pixel in my Direct Solar Radiation Raster. I can see the extent of the data in my raster but don't know where the highest value pertains to.   Is there a way to simply select pixel with highest value or create a point at highest value?