# Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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### Prove: if x is odd, then sqrt(x) is odd.

If $x$ is odd, then $\sqrt{x}$ is odd, where $x$ is an integer. Any hints welcome and preferred. Thank you!
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### How to show that $P_n(x)$ have n distinct roots [duplicate]

Let $P_n : \mathbb R \to \mathbb R$ , $n\in \mathbb N$ be defined by $P_n(x)=\frac{\displaystyle1}{\displaystyle2^n n!}\frac{\displaystyle d^n}{\displaystyle dx^n}[(x^2-1)^n]$ I need to show that ...
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### Converse of DIC

I am looking to prove the converse of the Divisibility of Integer Combination. I know how to prove the contrapositive of this statement but not the converse ... any help? Below is my attempt at ...
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### Question about $E(|Z|)$ at Normal distribution

$Z$ is a standard normal variable. How do I calculate $E(|Z|)$? ($E(Z)=0$). Thank you!
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### What's the fastest way to determine Eigenvalues & Eigenvectors of any 2 by 2 Matrix?

My instructor claims that it's inefficient and superfluous to compute eigenvectors de novo for each $2$ by $2$ matrix. He suggested a trick instead which resembles the eigenvectors and cases here. ...
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### How to Complete Sketch of a function of two variables $f(x, y) = 3x - x^3 - 2y^2 + y^4$ ? [Stewart P930 Question 14.7.4]

For $f(x, y) = 3x - x^3 - 2y^2 + y^4$ $\implies$ $\partial_x f = 3 - 3x^2, \partial_y f = -4y + 4y^3$. Set both equations to 0 $\implies x = \pm$1 and $y = 0, \pm 1$. $1.$ To determine the ...
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### Sketch Saddle Point of a function of two variables $f(x, y) = 4 + x^3 + y^3 - 3xy$ [Stewart P930 Question 14.7.3]

As regards $f(x, y) = 4 + x^3 + y^3 - 3xy$, I computed that (0,0) is a saddle point, and (1,1) is a local minimum. So I'm not asking about this, and am asking only about sketching contours. $1.$ ...
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### Prove that these Sets Containing Infinitely Many Incompressible Strings Exist

We define a set $A$ to be special if: $$\liminf_{n \to \infty} \frac{|A^{\leq n}|}{n} = 0$$ I want to prove that there are special recursive sets that contain infinitely many incompressible strings. ...
$12.$ (a) Determine all critical points of $\dfrac{dx}{dt}=(1+x)\sin y$ , $\dfrac{dy}{dt}=1−x−\cos y$ . (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of ...
By Ramsey's theorem we know that: $\forall k \in \mathbb N : \exists N \in \mathbb N$ that an arbitrary graph $G$ on a set of vertices $\{1,2,...,N\}$ contains $k$ vertices, which represent either a ...