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Questions tagged [extreme-value-theorem]

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Proof that the sum of two regularly varying functions at zero is regularly varying at zero?

I have given two r.v. $X$ and $Y$, both regularly varying at zero with exponent $\alpha$ i.e $$\lim\limits_{t \to 0} F(tx)/F(t)=x^\alpha,$$ where $F$ is the distribution function of $X$ ...
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2answers
21 views

Extreme value theorem: help with contradiction

I have a problem understanding the last part of the usual proof of the extreme value theorem (found for example here: Extreme Value Theorem proof help) It is this part that I have trouble ...
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0answers
23 views

What can we say about the ratio of two Minima/Maxima?

I have been reading my old script about extreme value theory as well as I've been searching the web, but I could not find any question like that; What I am wondering about: Assume we have given some ...
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1answer
37 views

Proving the maximum value of a set of numbers given their sum

Say we have $\sum_{i=1}^n x_n = C$, i.e., $x_1+x_2+...x_n=C$ where C is a constant and $x_1,x_2,...x_n$ are nonnegative. Prove the product $(x_1)(x_2)(x_3)...(x_n)$ has a maximum if and only if $x_1=...
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1answer
51 views

Proof of AM GM theorem using Lagrangian

Given: $\prod_{i=1}^n x_i = 1$ leads to constraint function $G(x_1,x_2,...,x_n)=\prod_{i=1}^n x_i-1$ ($\prod_{i=1}^n x_i =x_1 x_2...x_n$) Task is to to find the minimum using conditional extrema of ...
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Inproper Extreme Value Distribution of Type I (Gumbel) or badly estimated parameters?

I would appreciate a lead\verification on the following excercise. Let $W$ denote the anual maximum wind load. The anual maximum of $W$ is distributed according to an asymptotic extreme value ...
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1answer
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Find a max possible value of an undefined function in an interval when derivative is a constant [closed]

The function $f(x)$ is continuous and differentiable in $[0,1]$ if $f'(x)\le 10$ for all $x\in[0,1]$ and $f(0)=0$, What is the maximum possible value of $f(x)$ for $x\in [0,1]$ ? Any help would be ...
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2answers
37 views

Show that if f is continuous and periodic then f attains both its minimum and its maximum.

Show that if f is continuous and periodic then f attains both its minimum and its maximum. The solution is given below: But I wonder why he choose the k like this and is the solution does not ...
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1answer
34 views

Absolute conditional minimum of function in n-dimensional space

Function $$F(x_1,x_2,...,x_n) = \sum_{i=1}^n x_i$$ on the constraint $$G(x_1,x_2,...,x_n)=\prod_{i=1}^n x_i-1$$ what I come up with is writing down the Lagrangian: $$L = F(x_1,x_2,...,x_n) - \...
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2answers
214 views

Trouble understanding how the Transfer Principle is applied for the Extreme Value theorem.

I am reading Keisler's Elementary Calculus (which can be downloaded here). I am having trouble understanding his proof sketch of Extreme Value Theorem and how he is applying the Transfer Principle. ...
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1answer
31 views

Proving and disproving some statements based on Extreme Value Theorem

Here are the statements: Suppose that $f$ is continuous on the interval $[a,b)$ and that $\lim_{x\to b^{-}} f(x) = +\infty$. Then $f(a) \le f(x)$ for all $x$ in $[a,b)$. Suppose that $f$ is ...
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2answers
67 views

Generalized Way of Treating Extrema under Certain Constraints (Inequalities)

Let's take a simple example $f: \mathbb R^{2} \to \mathbb R$, $f(x,y)=xy$ and then I want to treat $f$ for a constraint $M$ under all possible inequalities: Case 1) $M:=\{(x,y)\in \mathbb R^{2}|x^2+y^...
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4answers
66 views

Difficulty finding Lagrange multiplier because of $\leq$

Let $f: \mathbb R^3 \to \mathbb R$ be defined by $$f(x,y,z)=x-y+z$$ and $$E:=\{(x,y,z)\in \mathbb R^{3} \mid x^2+2y^2+2z^2\leq1\}$$ Find the extrema of $f$ on $E$. Path: I have already proven that ...
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2answers
40 views

Show the existence of a global maximum of a continuous function with unbounded domain

I am given a function $f(t) \in \mathbb{R}$ which is continuous; bounded above by $M$ and below by $0$. $f$ is differentiable everywhere except at $f=0$. Also, $\lim_{t \to \infty} f = 0$ and $t \in [...
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2answers
39 views

Find out if a function has a maximum and minimum or not

$\psi: \mathbb{R}$ → $\mathbb{R}$ is a continuous function such that $\lim_{x\to +∞} ψ(x) = +\infty $ and $\lim_{x\to -∞} ψ(x) =-\infty $ Decide if G: $\mathbb{R}$ → $\mathbb{R}$, $G(x) = \frac{ψ(x)}{...
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1answer
67 views

What does it mean centering a Gumbel distribution?

Consider $M$ i.i.d. random variables $V_1,..., V_M$ distributed as Gumbel with location $\lambda$ and scale $\beta$. We know that (see proof at the end of the question) $$ E(\max_{k\in \mathcal{Y}} ...
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0answers
58 views

Extreme value theory - proof this is a poisson point process

Let $(X_n)_{n \geq 1}$ be an i.i.d sequence of real valued RVs with continous distribution function f and $M_n:=\max \{X_1,...,X_n \}$. Let $U_n:=\inf \{ k \in \mathbb{N} | X_k>X_{U_{n-1}} \}$ be ...
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0answers
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Extreme value index of $F(x) = 1-e^ \left(-\frac{x^2}{4}\right)$

I am asked to show that $F(x) = 1-e^{(-\frac{x^2}{4})}$ is in a max domain of attraction and to calculate the extreme value index. A typical question is solved using $lim_{t\rightarrow \infty}\frac{U(...
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2answers
51 views

Proving there must exist a maximum value in a continuous interval

I know this is part of the Extreme Value Theorem but I want to tackle this one bit first, focusing on the maximum case. For a function $f$ continuous over interval $[a,b]$, it has a max value over ...
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Statistics of Extreme risks

$F$ belongs to the maximum domain of attraction of the Frechet distribution $G_{1,\alpha}$: for some $\alpha > 0$, if and only if $x^{\alpha} \bar{F}(x) = L(x)$ is a slowly varying junction. The ...
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1answer
101 views

Generalized Pareto distribution (GPD)

I'm trying to understand the functional form of the Generalized Pareto distribution (GPD) presented in Wikipedia. In the "Definition" section location parameter $\mu$ does not appear in the function, ...
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2answers
47 views

Minimum distance from the points of the function $\frac{1}{4xy}$ to the point $(0, 0, 0)$

I am trying to find the minimum distance from the points of the function $\large{\frac{1}{4xy}}$ to the point $(0, 0, 0)$. This appears to be a problem of Lagrange in which my condition: $C(x,y,z) = ...
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0answers
32 views

Critical point of a multivariable function

For the function $f(x,y)=x^2y^2 + \frac{1}{y} + \ln(\frac{1}{x})$ I get two critical points, namely $P_1 =\left(\sqrt{\frac{1}{2}},1 \right)$ and $P_2 =\left(-\sqrt{\frac{1}{2}},1 \right)$. However ...
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3answers
45 views

Extreme values $f(x)=(x-2)^{\frac{1}{3}}$

find all points of intrest for the function: $f(x)=(x-2)^{\frac{1}{3}}$ Here we can clearly see that when $x=2$ $f(x)=0$ so I know that there atleast should exist a critical point. Since the ...
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5answers
1k views

Finding extreme values where second derivative is zero

Consider this function: $$f(x)= 5x^6 - 18x^5 + 15x^4 - 10$$ I am told to find the extreme values of this function. So at first, I take the first derivative and set it zero. $$f'(x)=30x^5-90x^4+60x^...
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2answers
205 views

Extreme values for a vector equation

For a question on physics.stackexchange about Does the Ampère-Maxwell law fail for the field of a uniformly moving point charge? with $$ \vec B(P) = \dfrac{\mu_0 q}{4 \pi} \dfrac{1 - v^2/c^2}{[1 - (v^...
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2answers
122 views

If a function is defined on a closed interval $[a,b]$, does it necessarily achieve a max and min value on that interval?

The extreme value theorem requires that a function be continuous on a closed interval $[a,b]$ for it to necessarily take on a max and min, but I've been thinking and it seems to me that as long as it ...
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1answer
49 views

Absolute Extremes of: $f(x,y,z) = xyz$ with $x+y+z=1$

I am attempting to find the absolute extremes of the function: $$f(x,y,z) = xyz$$ with the condition that: $$x+y+z=1$$ So far I have gathered the following: Condition: $$C(x,y,z) = x+y+z-1$$ and ...
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2answers
37 views

Determine the Conditional Extremes of a Function

I am trying to determine the conditional extremes the following question: Determine the point of the plane, $2x-y+2z=16$ closest to the origin. but I do not fully understand the question. If I am ...
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1answer
33 views

Absolute extremes of: $f(x,y)=x^3+xy+y$ in the enclosed triangle region limited by the lines $x=-1$, $y=3$, $y=x+2$.

I am trying to find the absolute extremes of the function: $f(x,y)=x^3+xy+y$ in the enclosed triangle region limited by the lines $x=-1$, $y=3$, $y=x+2$. So far I have been able to graph the triangle ...
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1answer
197 views

What are the points on a Return Level Plot?

A pretty simple question I think, but I can't seem to find an answer anywhere. I have produced the below return level plots for a project I am completing on stock market data. I understand the meaning ...
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2answers
320 views

Prove that a function attains its minimum

Let $f$ be a real continuous function defined on $D=[0,+\infty)$, $f(x)\geq 0$ for all $x\in D$, and $\lim_{x\to+\infty}\:f(x)=+\infty$. Prove that $f(x)$ attains its minimum on $D$. Idea for a ...
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3answers
49 views

Show $f(x,y) = y^2 - x^2$ at $(0,0)$ has a critical point, but is not a max/min value

So as always... I found the partial derivative with respect to $x$ and $y$ of $f(x,y)$ which gave me: $f_x=-2x$ $f_y=2y$ So I wasn't too sure what to do next, but I set $f_x = 0$: $0 = -2x$ $x=0$ ...
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1answer
50 views

Find maximum and minimum value of function of three variables on the set $E$

$$f(x,y,z)=4x+2y+z$$ $$ E=\{(x,y,z) \in R : (x+1)^2+4y^2+4z^2=4\}$$ I know I should write here what I already did but I could come up with literally nothing. Should I just find extreme values of $g(x,...
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1answer
53 views

Local extremes of: $f(x,y) = (x^2 + 3y^2)e^{-x^2-y^2}$

I am looking to find the local extremes of the following function: $$f(x,y) = (x^2 + 3y^2)e^{-x^2-y^2}$$ What have I tried so far? Calculate the partial derivatives: $$\frac{\partial f}{\partial x}...
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2answers
52 views

if $A,B \subseteq \Bbb R^n, A \cap B = \emptyset$ , $A$ compact and $B$ closed then the distance is achieved.

For 2 sets $A,B \subseteq \Bbb R^n$ such that $A \cap B$ = $\emptyset$, denote: $d(A,B):=$ inf$\{||x-y|| : x\in A, y \in B \}$. Show that if $A$ is compact and $B$ is closed, then there exists $a \...
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1answer
523 views

Proving that any continuous complex function is bounded on a closed bounded set

Let $E$ be a closed, bounded set and let $f(z)$ be a continuous complex function in $E$. Prove that $f(z)$ is bounded in $E$. I began the argument the same way that the boundness theorem is addressed ...
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1answer
49 views

Show that f has a minimun

been trying to solve this for some time now. f is continuous in $ [0,\infty), $ and $\lim_{x\to \infty}f(x) = L . $ prove that if there exist $x \ge 0 $ such that f(x) < L then f has a minimum ...
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1answer
53 views

Several questions about continuous, derivative and extrema

Those problems come with my proof of question. I already found a better solution for this question, but there exists some confusion in the first proof occur to my head Original Question f(x) is ...
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1answer
36 views

Extremum of a sum of polynomial and square root of polynomial

Let $f(x)$ be of the form $f(x) = P_1(x)+\sqrt{P_2(x)}$, where $P_1(x)$ is a monomial $P_1(x)=ax+b$ and $P_2(x)$ is a quadratic function $P_2(x)=cx^2+dx+d$, defined on the closed interval $[0,1]$. ...
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2answers
85 views

Extreme value theorem for $f:\mathbb{R}^n\to \mathbb{R}^m$

A marker comments that the EVT only considers functions of the form $f:\mathbb{R}^n\to \mathbb{R}$. However, I don't understand why this should be the case. For there is, for example, the notion of a ...
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$C^1$ function on compact set is Lipschitz

Let $\emptyset\ne A\subset\mathbb{R}^n$ be open and let $f \in C^1 (A, \mathbb{R})$ be a function. Let $\emptyset\ne K\subset A$ be compact and convex. I want to prove that $f$ is Lipschitz on $K$; ...
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3answers
337 views

Find extreme values of an implicit function

The prompt is to find the extreme values of an implicit function $z(x, y)$ The functions are $$ x^2 + y^2 + z^2 -3z = 0$$ $$x^2 + y^2 +z^2 -2x -2z +2 =0$$ Solving functions with just 2 variables, ...
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1answer
49 views

Finding the maximum and minimum of $f(x,y)= y(1-x^2-y^2)$ on $D:=${$(x,y)|x^2+y^2 \leq 1$} with extreme value theorem

Define $f : \Bbb R^2 \to \Bbb R$ by $f(x,y)= y(1-x^2-y^2)$ Let $D:=${$(x,y)|x^2+y^2 \leq 1$}. Does $f$ take a maximum and a minimum on $D$? If so in which points? So I calculated the critical points,...
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1answer
124 views

Applying Extreme Value Theorem to prove existence of unique fixed point

Let $K$ be a non-empty compact subset of $\mathbb{R}^n$, and let $f:K \to K$ be a function which satisfies that $\|f(x) - f(y)\|<\|x - y \|$ for all $x,y\in K$ where $x \neq y$. I want to prove ...
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0answers
31 views

Rate of convergence of the maxium of a random sequence

I came across a problem which requires the rate of convergence of $\sup_{1\le j \le N} |\sum_{i=1}^{N}X_{i,j}/N|$. If sup over a finite number of objects, this is a simple application of LLN. However, ...
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2answers
305 views

What is the critical points of $f(x,y) = e^{\sin x\cos y} $?

I try to find local extreme values and saddle point(s) of the $f(x,y) = e^{\sin x\cos y} $. But, when I take the partial derivative, I can't figure out to find critical points. $$f(x,y) = e^{\sin x\...
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2answers
290 views

Intermediate value theorem on infinite interval $\mathbb{R}$

I have a continious function $f$ that is strictly increasing. And a continious function $g$ that is strictly decreasing. How to I rigorously prove that $f(x)=g(x)$ has a unique solution? Intuitively, ...
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2answers
28 views

For what values of parameter m the function $g(x) - 2x^3 - 3x^2 + mx + 3 $ has an extremum of 10? An easier way to solve it

My attempt to solve this problem is very tedious and I do not think it is the optimal method. $$g'(x) = 6x^2 -6x + m$$ $$g'(x) = 0 \Rightarrow x = \frac{3 \pm\sqrt{9-6m} }{6}$$Now, I need to subsitute ...
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0answers
58 views

Frechet distribution function compared to empirical data

So basically I am trying to evaluate VaR on the Tesla stock using a Block Maxima method. I.e. I assume that the worst weekly log returns follow a generalized extreme value distrubtion with the shape ...