All Questions
1,375
questions
-1
votes
1
answer
84
views
$f_{n}(x)= x^n+n\sqrt x-1 $. Prove that $a_{n+1} < a_{n}$
$f_{n}(x)= x^n+n\sqrt x-1$
We already proved the following:
$f_{n+1}(a_n)= a_n^{n+1} + \sqrt{a_n} \cdot(1-a_n^{n-1}\sqrt{a_n}) $
$a_{n}$ is a solution to $f_{n}(x)=0$
$0 < a_{n} < 1$
1- Prove ...
- 7,748
1
vote
0
answers
93
views
Is there a sequence of $C^2$ functions approximating $\max(0,x)$?
I know that the sequence
\begin{equation*}
f_n(x):=\begin{cases}\sqrt{x^2+\frac{1}{n^2}}-\frac{1}{n},& x\ge0, \\\\ 0,&\text{otherwise},\end{cases}
\end{equation*}
is a $C^1$ sequence pointwise ...
0
votes
1
answer
28
views
function defined on set of either finite or complementary finite subsets of infinitely coutable universal set
Let $X:=\{x_i\}_{i=1}^\infty$ and let $F:=\{A\subseteq X|\text{ $A$ is finite } \lor\text{ $A'$ is finite } \}$; define a function $\mu:F\mapsto [0,\infty)$ a such that
$\mu:=\begin{cases} 0 \text{ | ...
- 13.8k
1
vote
2
answers
119
views
Can Big $O$ ever be negative? How does those concepts relate to the limit definition?
So for some time I am trying to figure out big $O$ and small $o$ notation and I understand the main idea behind those notations but when I try to dig a bit deeper it seems I can't figure some stuff, ...
- 3,105
0
votes
1
answer
63
views
Continuous non-vanishing vector field on annulus shape regions
Let $\Omega \subset \mathbb{R}^2$ be an open bounded region and $\Omega_0 \subset \subset \Omega$. Assume also that $\Omega_0$ is simply-connected. Suppose a continuous non-vanishing vector field $F$ ...
- 5,153
0
votes
0
answers
45
views
Is there an injective function which has the IVP but is not strictly monotonic? [duplicate]
Can we find an injective function that shows IVP but is not strictly monotonic?
I know that there are functions that are injective but not strictly monotonic, like $f(x)=\frac{1}{x}$ if $x≠0$ and
$f(...
- 39.7k
1
vote
0
answers
28
views
Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$
Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$
I would like to prove (or disprove) ...
- 24.2k
6
votes
2
answers
399
views
How prove $f(x)$ is monotonous , if $f'(x)=g[f(x)]$
Question:
Let $f(x)$ be a derivative, and there exsit $g(x)$ be such that:
$$f'(x)=g[f(x)]$$
Show that $f(x)$ is monotonic.
This problem is from Xie Hui Min analysis problems book in china ,...
3
votes
6
answers
362
views
Local minimum global
Let $f:(a,b)\to\Bbb R$ be continuous. Assume that $f$ has a local minimum at some point $x_0$. Further assume that this is the only point where $f$ has a local extremum. Does it follow that $f$ has a ...
- 832
1
vote
0
answers
40
views
On subadditive functions everywhere finite bounded on compact sets
In the book "Functional analysis and Semi-Groups" by E Hille and R S Phillips, theorem 7.4.1 states that subadditive functions defined on some interval $I$ and finite everywhere are bounded ...
- 40.7k
0
votes
2
answers
41
views
Determining maximum and minimum of the function
Determine the minimum and maximum of the function $f(x,y) = xy$ in the annulus $A = \{(x,y) \mid 1 \leq x^2 + y^2 \leq 4\}$.
I'm wondering if in cases like this, where we have two conditions for the ...
- 77k
2
votes
1
answer
126
views
To show the given limit does not exist
Given the following function
$$f(x)=2 x \sin \left(\frac{1}{x}\right)-\cos \left(\frac{1}{x}\right) ~~ \text{for} ~~ x \neq 0 ,$$
to show that $\displaystyle \lim _{x \rightarrow 0} f(x)$ not exist, I ...
- 3,051
4
votes
1
answer
538
views
Intuition behind Legendre convex function
I came across the definition of Legendre functions and Legendre transformations in my studies (in the sense of convex analysis) and I started searching about it.
I found a definition in Rockefellar's ...
- 4,878
0
votes
0
answers
54
views
Show that the polynomial function $f(m,n)=1/2((m+n)^2+3n+m)$ is one to one and onto [duplicate]
I'm trying to prove that $\mathbb{N}\times\mathbb{N}$ is equinumerous to $\mathbb{N}$ using that fact the function $f:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ defined by $f(m,n)=1/2((m+n)^2+3n+...
- 1,289
-2
votes
2
answers
112
views
How do I prove interval 𝐴⊂[0,3] exists on this integration
Let $f:[0,3] \to \mathbb{R}$ be a continuous function satisfying
$$\int_{0}^{3}x^{k}f(x)dx=0 \quad \text{for each k = 0,1,}\dots,n-1$$ and
$$\int_{0}^{3}x^{n}f(x)dx=3.$$
Then prove that there is an ...
- 1,202