I just got a little lost in the assumptions, I first assumed that $~y&gt; x~$, but then I wasn't very successful.
122205
I think I have proved that that a cannot be greater than b (ie a &amp; b must be equal) then using this fact I got a quadratic in terms of a and c, trying out some values I got that (3,3,4) is a solution, but how do I prove this is the only solution or if this is not the only solution then what are the other solutions and how do I find them?
45509
I note from the archive discussion of the case for integers, which I'm fine with.   For the positive reals, I have set up axes for x, y and z = x ^ y. The solution set is then graphed by a space curve in the surface, necessarily symmetrical about the plane y = x. I need the equation of the orthogonal projection of this curve onto the x-y plane. (All I have shown so far is that  $x ^ y &gt; y ^ x$ in the region right of the line y = e and above the line y = x.)
49107
I can't seem to figure out the inequality option, I substituted n=2 for the answer bit is there any elegant way of getting it?
110506
Is there any counterexample to show that $X,Y$ are two random variables and $E(X\mid Y)=E(X)$, but $X$ and $Y$ are not independent.   I already know that if $X,Y$ are independent, then $E(X\mid Y)=E(X)$.
91932
How can I attack this problem?  My idea is for $X \subset \mathbb{N}$ set up a linear order $O_{X}$ such that if $X \not =Y$ then $O_X \not = O_Y$.
101867
I can't get any of these on my own and I am attempting to do $x^x$ I had it explained to me for about an hour and I still can't do it on my own. I thought I was supposed to make it into $x(\ln x)$ which should be equivalent to the original term. Then the differentiation should be easy for most people from here.  $$1(\ln x)+x(\frac1x) $$ or $$\ln x+1$$  This is of course wrong but I do not know why.
92444
I am having a little trouble understanding what I have to actually do here. What does differentiate with respect to bn? I thinks after differentiation I must use some calculus theorem about extreme but how do I get there?
35166
There are several proofs I have seen of this, but they all seem to use choice subtely at some point. Is there any way to prove this without choice, or is it possibly unproveable?
28630
Given that   $x^x = y$; and  given some value for $y$   is there a way to expressly solve that equation for $x$?
38272
I have been told it is very simple, but I cannot work it out. I was thinking I use that I know the left hand side converges?
67560
I am stuck on the 4 th one.I have done the rest.I have found out the value of a_n.But not getting how to proceed for the 4 th one.
97319
Can you please explain why \begin{align*} \lim_{x\to 0^+}{x^{x^x}} &amp;= 0 \end{align*}
47039
Just doing this for revision, seems much harder than it should be, should I use $x=\tan u$ ? Any help appreciated.
122977
I have to prove that the result of $x * y \in G$ so $\frac{x+y}{1+xy} \in (-1, 1)$. So $x &gt; -1$ and $y &gt; -1$ at the same time $x &lt; 1$ and $y &lt; 1$.  If I multiply the first 2 expressions I obtain $xy &lt; 1$ which is true, and if I sum up the last 2 I get $x + y &lt; 2$. At this point I am not sure if I am doing well. How should I continue this problem?
68436
I'm trying to integrate by parts but it has not been very effective.
89364
I've found for my econometrics exams that if I forget the scalar notation, I can often save myself by remembering the matrix notation and working backwards. However, the following confused me.  Given the simple estimation  $$\hat{y_i} = \hat{\beta_0} + \hat{\beta_1}x_{i1}$$  How do we get from  $$\boldsymbol{\hat{\beta}} = \boldsymbol{(X'X)}^{-1}\boldsymbol{X'y} $$  to  $$\hat{\beta}_1 = \frac{\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^n(x_i-\bar{x})^2} $$  I get stuck at  $$\hat{\beta}_1 = \frac{\sum_{i=1}^nx_iy_i}{\sum_{i=1}^nx_i^2}$$
19379
It is stated as a problem in Spivak's Calculus and I can't wrap my head around it.