Questions tagged [extreme-value-theorem]

This tag is for questions relating to "Extreme Value Theorem". The Extreme Value Theorem guarantees both a maximum and minimum value for a function under certain conditions.

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13 views

Stationary points problem

I already made a first derivation of $f\left ( s,t \right )$. For $\frac{\partial f}{\partial s}=4s^{3}-2s-2t$ and for $\frac{\partial f}{\partial t}=4t^{3}-2s-2t$. I have to find the stationary ...
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1answer
32 views

Local extreme of function : $f\left ( x,y \right) =x^{2}+2x+y^{2}-2y+3$ [closed]

I am supposed to find the local extreme of function : $f\left ( x,y \right) =x^{2}+2x+y^{2}-2y+3$ on circle with the origin in the coordinate axis and radius $2\sqrt{2}$. I know how should I do this, ...
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1answer
34 views

What is the maximum possible value for $f(x)$ for $x \in [0,1]$? [closed]

A 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 $[0, 1]$?
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1answer
54 views

Second derivative test in the Hilbert space case

Let $H,E$ be $\mathbb R$-Hilbert spaces; $f\in C^2(\Omega)$; $c\in C^2(\Omega,E)$; $M:=\left\{c=0\right\}$; $x\in M$ be a local minimum of $f$ constrained on $M$, i.e. $$f(x)\le f(y)\;\;\;\text{for ...
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2answers
38 views

Find the minimum value of the function a(x).

$$a(x)= \sqrt{x^3}+\sqrt{x^{-3}}-4(x+\frac{1}{x})$$ One of the ways I could think of was to find out the global extreme values and proceed.But as I began doing it that it takes a lot of ...
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2answers
35 views

Example of a calculus optimization problem where the answer occurs at an endpoint

I'm teaching optimization problems in calculus right now. An easy example would be something like: Find the dimensions of a rectangle with perimeter $100$ m whose area is as large as possible. The ...
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1answer
22 views

Proof Verification for Extreme Value Theorem

I had an idea for a proof of the Extreme Value Theorem, and I was wondering if it was valid. Any advice you have would be greatly appreciated. Thank you! Prove: If a function f(x) is continuous on a ...
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11 views

How to fully estimate a probability density from only a sample of minimum values?

We are given a sample $\{ z_i \}$, $i=1,2,\ldots,N$, such that each value $z_i$ corresponds to the minimum of $n$ random variables $x$, i.e., $z = \min \{ x_1, x_2,\ldots,x_n \}$. By means of ...
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0answers
36 views

If $u$ is an extremal point of a functional $E$ and $γ$ is a smooth curve with $γ(0)=u$, then $0$ is an extremal point of $E\circ\gamma$

Let $X$ be a $\mathbb R$-Banach space, $M\subseteq X$ be open, $M_0\subseteq M$ be closed, $E\in C^1(M)$, $u\in M_0$ be an extremal point of $E$, $\varepsilon_0>0$ and $\gamma\in C^1(-\varepsilon_0,...
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43 views

Probability that a normal random variable is greater than the maximum of $n$ i.i.d. random variables

Let $M_n = \max\{X_1,\ldots,X_n\}$, where $X_1,\ldots,X_n$ are i.i.d. random variables. We know about $M_{n}$ from the extreme value theorem and know it's mean and variance. Let $Y \sim \mathcal{N}(\...
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2answers
64 views

Open sets in a particular norm.

Let $f \in C^2[0,1]$, define $||f||_{2,\infty} := \underset{x \in [0,1]} {\text{sup}} |f(x)|+ \underset{x \in [0,1]} {\text{sup}}|f'(x)|+ \underset{x \in [0,1]} {\text{sup}}|f''(x)|$. Then I wish to ...
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0answers
34 views

Asymptotic behavior of the maximum of independent but non-identically distributed Gaussian random variables

Is there any result on the asymptotic behavior of the maximum of independent but non-identically distributed Gaussian random variables? Something similar to the result by Gnedenko, (1947) that for a ...
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23 views

Higher order derivative and extrema, theorem.

I wanted ask, maybe someone here can give a site or other source as to where could I find a theorem about higher order derivatives. The conclusion of theorem is as follows: If $f'(x_0)=f''(x_0)=f'''(...
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16 views

Extreme value theory Insurance

I've got data-set which is very large (16 million + ) and has over 150 covariates (which some are hot-encoded). This data-set spans across roughly 10 years These covariate inlcude details of the ...
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1answer
86 views

Show that a continuous periodic function on $\mathbb{R}$ attains its maximum and minimum. - Proof Verification

Let $f$ be a continuous periodic function on $\mathbb{R}$, that is $\exists\ d > 0$ s.t $f(x+d) = f(x)\ \forall x \in \mathbb{R}$. I believe I have the whole idea of how to prove this, but I'm ...
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0answers
17 views

Tail of CDF of non-central chi squared RV using asymptotics of Bessel function

The pdf and cdf of the non-central chi squared RV (under the scenario I am studying) is given as follows: \begin{align} &f(x)=\frac{1}{v} \exp\left(\frac{-(a+x)}{v}\right)I_{0}\left(\frac{\sqrt{xa}...
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1answer
54 views

How to find extreme values of an $f(x,y)$ function?

I need this for my semester exams, unfortunately I was absent the day this topic was "talked about". My function is the real-valued $$f(x,y)=x-xy+x^2+y^2$$, interpreted on $\mathbb{R}^2$. Single-...
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1answer
148 views

Derivation of Gumbel Distribution

The standard generalised extreme value (GEV) distribution is given by $H_{\xi}$ which is $exp(-(1+\xi x)^{-1/\xi}$ if $\xi<>0$ and $exp(-e^{-x})$ if $\xi=0$ In the lecture notes it is stated ...
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1answer
268 views

Let $f(x)$ be a continuous function on $[0,1]$ and $f(0) = f(1)$. Let $α ∈ (0,1)$. Prove that there exists an $x ∈ (0,1)$ such that $f(x) = f(αx)$.

Suppose $f(x)$ is a continuous function on $[0,1]$ with $f(0) = f(1)$. Let $α ∈ (0,1)$. Prove that there exists an $x ∈ (0,1)$ such that $f(x) = f(αx)$. I tried $h(x) = f(x) - f(αx)$ and Intermediate ...
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2answers
67 views

How do we construct a continuous function on the interval $(0, 1]$ without a minimum or a maximum?

One of my Calculus lecture videos poses the following challenge (shortly after an exposition of the Extreme Value Theorem): Construct a function $f$ such that $f$ is continuous on $(0, 1]$ ...
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1answer
40 views

I need to clarify my understanding of the extreme value theorem.

I have had some difficulty in trying to understand the extreme value theorem but I think I might understand it correctly now but would like to clarify that I am thinking about this correctly. If we ...
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1answer
32 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 ...
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1answer
26 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 $f(x,y)$ ...
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10 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. ...
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20 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 ...
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2answers
70 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 $a>0$.
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1answer
64 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.
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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.
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1answer
52 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 $f(t) = M$. ...
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0answers
17 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-...
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0answers
21 views

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

Show there is a point of the plane $\{x \in \mathbb{R^3} \mid x_1 + 2x_2 + 3x_3 = 13\} = A$ closest to the point $(1, 1, 1)$. Let $B[(1, 1, 1), r]$. Let a function $f: A\cap B \rightarrow \mathbb{R}$...
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0answers
56 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}...
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3answers
56 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(...
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2answers
53 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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1answer
38 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
69 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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1answer
23 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 ...
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2answers
142 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
38 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
274 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
40 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
74 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
81 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
83 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
51 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
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1answer
109 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
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0answers
88 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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2answers
78 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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1answer
173 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
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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) = ...