All Questions
30
questions
2
votes
0
answers
49
views
Prove that $(f+g)(x)<t$ ($t\in\mathbb{R}$) holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$.
I got stuck on this question:
Prove that $(f+g)(x)<t$, $t\in\mathbb{R}$, holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$.
I think one direction is ...
- 2,473
3
votes
1
answer
136
views
Proving that $f(x)$ satisfying $2f(x)f(1)\leq f(x)f(1)+1\leq 2f(2x)$ for all $x>0$ is a constant function
How do I prove the following, where $f:[0,\infty)\to[0,\infty)$:
For $x>0$ if $f(x)$ satisfies
$$2f(x)f(1)\leq f(x)f(1)+1\leq 2f(2x)$$
then $f(x)$ is a constant function.
I have out found out ...
- 45
2
votes
2
answers
93
views
Alternative proofs of this Inequality
So I was reading a paper which made the claim "It is easy to see that $\frac{1-e^{-\alpha}}{\alpha} > 1-\frac{\alpha}{2} > \frac{1}{1+\alpha}$ when $0 < \alpha < 1$."
Verifying ...
- 5,880
2
votes
0
answers
94
views
Which assumptions on $f$ and $T$ make $\int_0^T f(x) dx \le \int_0^T |f(x)|^2 dx\implies 1\le \int_0^T |f(x)| dx$ true?
Let $T>0$ be fixed and let $f$ be a real valued function such that
$$\|f\|_{\infty}\le T,$$
where $\|\cdot\|_{\infty}$ denotes the sup-norm.
My question is the following: if
$$\int_0^T f(x) dx \le \...
- 3,277
0
votes
0
answers
51
views
Gravitational potential energy (explanation)
Hello to you dear person reading, I need your help to please explain me this :
$${\displaystyle \Phi (P)={\frac {-GmM}{\|OP\|}}+K}{\displaystyle \Phi (P)={\frac {-GmM}{\|OP\|}}+K}$$
(I haven't learned ...
- 1
1
vote
1
answer
57
views
Property on the infimum
Let $f: I \to \mathbb{R}$ be a function, where $I \subset \mathbb{R}^+.$ Let $u>0,E=\{x \in I,f(x)>u\}$ . Define $y \in \overline{\mathbb{R}}^+:y=\inf E$ if $E \neq \emptyset$ and $y=+\infty$ ...
- 11
0
votes
1
answer
52
views
An inequality for an increasing, concave function
Let $f:[0,\infty)\rightarrow[0,\infty)$ be a smooth function such that $f(0)=0$,
$f''<0$, $f'(x)>0$ for $x\in(0,N)$ and $f'(N)=0$. I would like to show that
$$
\max_{x\in(0,\infty)}\bigg(\sum_{...
- 797
0
votes
1
answer
128
views
Alternative proof of $x^x \geq \sin x$ if $x>0$
My lecturer gave me this exercise: "Show that for $x>0$ it is $x^x \geq \sin x$."
This is my approach: since $x \geq \sin x$ for all $x \geq 0$, it is sufficient to show that $x^x \geq x$;...
- 2,162
3
votes
1
answer
115
views
Proving or disproving: If $0<a<b<1$, then $(1-a)^b>(1-b)^a$
Prove or disprove
If $0<a<b<1$, then $$(1-a)^b>(1-b)^a$$
I think this looks true when evaluating the differential equation $\frac{dy}{dx}=-y$ with initial condition $y(0)=1$ using euler ...
- 305
0
votes
1
answer
61
views
Show that the function $f(x)=\frac{1}{x}-\left(\frac{d}{2x+2}\right)^d$ is always positive for $x\geq 2$ and $d \in \{2,3,4,5\}$?
My first instinct was to show that the derivative is always negative for the conditions above and with the limit $=0$ it should follow that it is always positive but it is way to complicated for me. ...
- 1
1
vote
2
answers
197
views
A function inequality about $e^x$ and $\ln x$
If for any $x \in (1,+\infty)$，there is the inequality:
$$x^{-3} e^{x}-a \ln x \geq x+1$$
Find the value range of $a$ .
And I tried constructing the function $f(x)=x^{-3} e^{x}-a \ln x - x-1$ and ...
- 125
3
votes
3
answers
131
views
$x-\sin(x) \geq \dfrac{x^3}{(x+\pi)^2}$
Let $x \geq 0.$ I need to prove that $x-\sin(x)\geq\dfrac{x^3}{(\pi+x)^2}.$
I tried the derivative, of $f(x)=x-\sin(x)-\dfrac{x^3}{(\pi+x)^2}$ which is $1-\cos(x)-\dfrac{x^2(x+3\pi)}{(\pi+x)^3},$ but ...
- 1,070
4
votes
2
answers
298
views
Monotonicity of function averages
Please let me know if you know an answer to this problem. May be you could provide a reference to some publication on this topic?
Let $f(x)$ be a real-valued strictly convex function on $[0, 1]$. ...
- 41
5
votes
2
answers
232
views
For continuous, monotonically-increasing $f$ with $f(0)=0$ and $f(1)=1$, prove $\sum_{k=1}^{10}f(k/10)+f^{-1}(k/10)\leq 99/10$
A question from Leningrad Mathematical Olympiad 1991:
Let $f$ be continuous and monotonically increasing, with $f(0)=0$ and $f(1)=1$.
Prove that:$$
\text{f}\left( \frac{1}{10} \right) +\text{f}\...
- 873
2
votes
1
answer
100
views
Connection between relationships of the form $\|f \|_p \leq \|g\|_p$
Let us define the $L^p$ norm of a function $f:\mathbb{R} \to \mathbb{R}$ as
$$
\|f\|_p:= \left(\int \lvert f \rvert^p\right)^{1/p}.
$$
Let us label the following statement by $S\left(p\right)$:
$$
\|...
- 400