Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [extreme-value-theorem]

The tag has no usage guidance.

2
votes
1answer
30 views

Question about a certain part of Weierstrass extreme value theorem

The lemma says that if $A \in \mathbb{R}^{n}$ is compact and $f:A \to \mathbb{R}$ is continuous in $A$, then there's a real $M$ such that $\forall X \in A,f(X) \leq M$. My textbook's proof goes this ...
0
votes
1answer
23 views

Finding points which are not local maximum or minimum

Consider the picture below: This is the levels curves for a function $f(x,y)$ where: Blue line is the partial derivative of $f(x,y)$ with respect to x Red line is the partial derivative of $f(x,y)$ ...
0
votes
0answers
8 views

How to determine if a critical point is a min/max on a function $f(x,y)$ on MATLAB

I am supposed, given a critical point [x,y] on a function f(x,y), determine if the point is a max/min without using the second derivative test. So I have a program which calculates crtical points. ...
0
votes
0answers
19 views

Creating variable functions using MATLAB

So I have three seperate function in MATLAB where each have its designated purpose. The first one calculates the partial derivative The second finds the roots for a system of two equations and two ...
2
votes
2answers
68 views

Can the ratio of the two smallest element of an iid sample converge to 1 if the support of $X$ is positive?

We have: $\mathbb P(X \leq 0)=0$ and $\mathbb P (X \leq a)>0$ for any $a>0$.
0
votes
1answer
46 views

Reference for extreme value theorem.

I look for a reference of the Extreme Value Theorem for semicontinuous functions defined on a topological space. I know the proof, but I want to cite this result in my work.
0
votes
2answers
26 views

Example of a function f and a set E with the following: f is uniformly continuous on E, but f doesn't attain either a max or a min on E.

I thought I could use a constant function but I think absolute maximums are attained on a constant function.
0
votes
1answer
24 views

Showing that the pointwise limit of continuous functions equals its supremum somewhere on compact domain

I'd appreciate hints on proving the following theorem: If $f(x) = \lim_{n\to\infty} f_n(x)$ for each $x \in [0,1]$ and $M = \sup_{x\in[0,1]} f(x)$, then there is $t \in [0,1]$ such that $f(t) = M$. ...
0
votes
0answers
10 views

Reverse Hoeffding Inequalities

Suppose that $X_t$ is a super-martingale, the Hoeffding inquality gives an exponential upper bound on the quatity $$ \mathbb{P}\left( \sup_{0 \leq t\leq T}X_t \geq x \right). $$ When can a lower-...
0
votes
0answers
15 views

Show that a point is closest to another point with extreme value theorem

Show there is a point of the plane $\{x \in \mathbb{R^3} \mid x_1 + 2x_2 + 3x_3 = 13\}$ closest to the point $(1, 1, 1)$. Let a function $f: A \rightarrow \mathbb{R}$ defined for all $x \in A$ by $f(...
1
vote
0answers
46 views

Domain of attraction $F(x)=\exp(-x-\sin(x))$

I need to show that $F(x)=\exp(-x-\sin(x)),x >0$ is in no domain of attraction. That means that there exist no $a_{k} >0, b_{k} \in \mathbb{R}, k \in \mathbb{N}$ with $\lim\limits_{k \to \infty}...
0
votes
3answers
35 views

Suppose a continuous function attains its minimum, prove that the function is not injective

Suppose a continuous function $f: (0,2) \to \mathbb R $ attains its minimum at $x_0 \in (0,2)$, prove that the function is not injective. We need to show there are some $a$ and $b$ such that $f(a)=f(...
0
votes
2answers
29 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 ...
0
votes
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=...
0
votes
1answer
60 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 ...
0
votes
0answers
11 views

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 ...
-1
votes
1answer
22 views

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 ...
1
vote
2answers
49 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 ...
2
votes
1answer
35 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) - \...
8
votes
2answers
233 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. ...
0
votes
1answer
32 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 ...
0
votes
2answers
71 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^...
1
vote
4answers
71 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 ...
1
vote
2answers
56 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 [...
0
votes
2answers
41 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)}{...
1
vote
1answer
85 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}} ...
3
votes
0answers
73 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 ...
0
votes
2answers
60 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 ...
0
votes
1answer
130 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, ...
3
votes
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) = ...
1
vote
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 ...
0
votes
3answers
46 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 ...
4
votes
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^...
1
vote
2answers
208 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^...
2
votes
2answers
320 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 ...
0
votes
1answer
54 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 ...
0
votes
2answers
41 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 ...
0
votes
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 ...
0
votes
1answer
373 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 ...
2
votes
2answers
386 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 ...
2
votes
3answers
50 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 $f_x = 0$: $0 = -2x$ $x=0$ ...
0
votes
1answer
55 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,...
1
vote
1answer
56 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}...
1
vote
2answers
58 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 \...
2
votes
1answer
801 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 ...
0
votes
1answer
50 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 ...
0
votes
1answer
54 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 ...
0
votes
1answer
37 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 $[0,1]$. ...
0
votes
2answers
99 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 ...
9
votes
2answers
2k views

$C^1$ 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 $K$; ...