# Tag Info

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### Question re: Collatz Conjecture

If we knew that every Collatz orbit eventually hits a number less than $10$ (is that what you mean?) then we'd be done immediately. So we don't know that! More generally if we knew that every Collatz ...
Accepted

### Let $x$ be a fixed real number. Prove: if $x^5 + 2x^3 + x < 50$, then $x < 2$

Note that $2^5+2\times2^3+2=50$, so for all $x>2$ you have $x^5+2\times x^3+x=50$. Can you conclude?
1 vote
Accepted

### Is there a way to define universal quantification in such a way, that I can prove$\forall xPx\land Q\iff \forall x(Px\land Q)$ and similar statements?

You can prove this equivalence this way over a finite domain, but there are other ways to this type of equivalence that work for infinite domains and are easier. I actually don't know whether the ...
1 vote

### Theorem 6, Section 2.3 of Hoffman and Kunze’s Linear Algebra

I think is false: In ${\Bbb R}^3$ with canonical basis let $e_1+e_2,e_2$ be a basis for $U$ and $e_1+e_3,e_3$ be a basis for $W$. Then $U\cap W$ is generated by $e_1$ which is not among the given ...
1 vote

### Let $x$ be a fixed real number. Prove: if $x^5 + 2x^3 + x < 50$, then $x < 2$

You can find the zeroes of $x^5 + 2x^3 + x$ using your factorization. For $x^4 + 2x^2 + 1$, you can solve $u^2 + 2u + 1 = 0$, where $u = x^2$. Once you solve for all the zeroes, you know that a ...
1 vote

### Let $x$ be a fixed real number. Prove: if $x^5 + 2x^3 + x < 50$, then $x < 2$

Let $f(x) = x^5 + 2x^3 + x$ So, $\frac{df}{dx} = 5x^4 + 6x^2 + 1$ Clearly, $\frac{df}{dx} > 0 \;\; \forall x \in \mathbb{R}$. So the function is strictly increasing. This means if $f(x_0) = c$ for ...

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