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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3
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1answer
49 views

How to use the extended version of Weierstrass's theorem?

After my question Question for the function $f(x)=\log\left(\frac{x^2}{x-2}\right)$, I have obtain a very good answer and I remember that I have never studied this theorem during my period at my ...
4
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1answer
83 views

Let $f: [a, b]\rightarrow R$ be differentiable at each point of $[a, b ]$ and $f'(a)=f'(b)$, prove that there's a line passing to $a$ tangent to $f$

Let $f: [a, b]\rightarrow R$ be differentiable at each point of $[a, b ]$, and suppose that $f'(a) = f'(b)$. Prove that there is at least one point $c$ in $(a,b)$ such that $$ f'(c) = \dfrac{f(c)-f(a)}...
1
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0answers
17 views

Find sufficent condition for $g(y)$

Let $x_0$ and $y_0$ - stationary points of twice differentiable functions $f(x)$ and $g(y)$. And $f(x_0)=0$. Find sufficent condition for $g(y)$ so that function $z(x,y)=f^2(x)\cdot g(y)$ has an ...
0
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0answers
14 views

Bound of the variance of maximum of normal random vaiable

I have searched related questions for a whole day, many posts are related but fail to solve my problems. If $X_1,,X_2\cdots,X_n$ ~ $N(0,\sigma^2)$, where $N(,)$ means gaussian random variable. I want ...
0
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0answers
19 views

Time until the next maximum?

Suppose that every year we sample $n$ random variables from a normal distribution. We want to keep track of the maximum value. On average, what is the meantime arrival between the maxima for the first ...
2
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0answers
24 views

Independence of asymptotic joint distribution of order statistic

Suppose $X_1,X_2,...,X_n\sim f(x)$ where $f(x)$ is a pdf and even function, i.e. $f(x)=f(-x)$. Now given that $$a_n(X_{(n:n)}-b_n)\xrightarrow{dist.} G_1$$ Then obviously from symmetry we can conclude ...
2
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6answers
60 views

Finding the absolute extrema of $F(x) = 2x + 5\cos(x)$ [closed]

Find the absolute extrema of $F(x) = 2x + 5\cos(x)$ on the interval $[0,2\pi]$ using the extreme value theorem. Answer should be 2 ordered pairs. I got $\arcsin(2/5)$ for the first value of $x$, but ...
9
votes
2answers
128 views

Given $f(x)$ is continuous on $[0,1]$ and $f(f(x))=1$ for $x\in[0,1]$. Prove that $\int_0^1 f(x)\,dx > \frac34$.

Let $f$ be a continuous function whose domain includes $[0,1]$, such that $0 \le f(x) \le 1$ for all $x \in [0,1]$, and such that $f(f(x)) = 1$ for all $x \in [0,1]$. Prove that $\int_0^1 f(x)\,dx >...
0
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2answers
16 views

Equivalence of EVT consequence

I was getting over EVT whics states that if a real-valued function $f$ is continuous on the closed interval $[a,b]$ then $f$ must attain a maximum and a minimum, each at least once. Then there is ...
0
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1answer
21 views

Find the critical points of the function $f(x,y)=(y-x^2)(y-2x^2)$ and examine if they are local extremums.

I have given a function $f(x,y)=(y-x^2)(y-2x^2)$. I have to inspect its critical points and to see if they are actually local extremums. I found the critical points by setting $ \nabla f(x,y)=0$ and ...
0
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2answers
35 views

Is Lagrange multipliers and (multivariable) extreme value theorem related?

I couldn't find a question answering this concept but they seem to be related. Extreme Value Theorem (two variables) If f is a continuous function defined on a closed and bounded set $A⊂\mathbb{R}^2$,...
1
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1answer
31 views

If $L(y)=\min_{x\in \mathbb{R}}h(x,y)$ is well defined in $(a,b]$ then is it also well defined on $a$?

Let $h:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be a continuous function such that $\lim_{x\to \pm\infty}h(x,y)=0$ for all $y$. Then we know that if $L(y)=\min_{x\in \mathbb{R}}h(x,y)$ exists, it ...
1
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1answer
22 views

Find the extreme values of an absolute function on a given interval

(Q) Find the extreme values for $f(x)=|3x-5|$ on $-3≤x≤2$ Because this is an absolute function $f'(x)=0$ does not exist. There is a local minimum at $x=\frac{5}{3}$ For the interval $-3≤x≤2$, $$f(-...
0
votes
1answer
11 views

Extreme value points of a polynomial.

Im only quite confident about quadratic equations and as far as I've heard, extreme values are maxima and minima. Also,I've heard that a polynomial of degree n has at most (n-1) extreme value points. ...
0
votes
1answer
19 views

Show that a function is continuous in order to use extreme value theorem

Let $f:\mathcal{Z}\times\mathcal{Z}\rightarrow \mathbb{R}$ be a function defined on a compact metric space $(\mathcal{Z},\rho)$ such that $\forall z_1,z_2\in\mathcal{Z}$ \begin{equation} \label{eq1} |...
1
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1answer
48 views

Distribution of a sequence of maximums generated using i.i.d. Normal variables

I am trying to think about the distribution of a random process. Here's how you would generate the sequence: for each sample of size k (sampled from i.i.d. Normal R.V.s), we find the maximum, and let'...
3
votes
1answer
102 views

Variance of max of $m$ i.i.d. random variables

I'm trying to verify if my analysis is correct or not. Suppose we have $m$ random variables $x_i$ , $i \in m$. Each $x_i \sim \mathcal{N}(0,\sigma^2)$. From extreme value theorem one can state $Y= ...
-1
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1answer
32 views

Understanding the statement of the Extreme Value Theorem

I am trying to understand this statement of the Extreme Value Theorem: If $f:K \rightarrow \mathbb{R}$ is continuous on a compact set $K \subseteq \mathbb{R}$, then $f$ attains a maximum and ...
0
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0answers
18 views

Spherical geometry (optimizing functioin)

I'm currently working on the following problem: Given points $H$ and ${X}_{1},\dots,{X}_{n}$ on a sphere and let ${d}_{i}$ denote the spherical distance between $H$ and ${X}_{i}$. Further let $f$ be ...
0
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1answer
31 views

Application of IVT to Proof of MVT for Integrals

I'm trying to understand the proof for the mean value theorem of integrals. I was looking at the following method of proof where after using the extreme value theorem to obtain a maximum and minimum ...
0
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1answer
17 views

Infimum and supremum of x/sinx in (0,π/2] [closed]

Please help I mean unable to do this even after using derivative technique
1
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2answers
27 views

Limiting distribution to Weibull

I am trying to solve the following question: Let $X_1,X_2,...$ i.i.d with $F(x) = P(X_i < x),~\\M_n := \max_{1\le{i}\le{n}}\ X_i,~\ $ $F(x_0) = 1$ and $F(x)<1$ for all x Given; $\lim_{x\...
0
votes
3answers
44 views

Finding local maxima and minima of function

I am supposed to find the local extreme of function $z=xy$ on set $x^2+y^2=1$. I used substitution:$ x=\cos \theta ,y=\sin \theta $, where $\theta \in \left [ 0,2\pi \right ]$. So: $z=\cos \...
0
votes
2answers
38 views

Finding local extrema

I am supposed to find local extrema of function: $f\left ( x,y \right > )=x^{2}+y^{2}-5xy$ on triangle ABC, where A=$\left ( 1,1 \right )$, B=$\left ( 3,2 \right )$, C=$\left ( 1,7 \right )$ I ...
1
vote
1answer
37 views

Extreme Value Theorem on an Unbounded Domain

Given a smooth function $f(x)\colon \mathbb{R} \to \mathbb{R} $. Suppose $$\lim_{x\to -\infty} f(x) = 0, $$ and $$ \lim_{x\to \infty} f(x) = 0. $$ Can we claim that $f$ is bounded?
0
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1answer
266 views

A continuous function on a compact set is bounded and attains a maximum and minimum: “complex version” of the extreme value theorem?

My textbook, Complex Analysis, by Shakarchi and Stein, gives the following theorem: Theorem 2.1 A continuous function on a compact set $\Omega$ is bounded and attains a maximum and minimum on $\Omega$...
0
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0answers
22 views

Parameter estimation for non-stationary GEV model with covariates

I’m trying to estimate the parameters for a non-stationary generalised extreme value (GEV) distribution (see below). $$Z_t \sim GEV \left( \mu (t) , \sigma, \xi \right)$$ $$\begin{split} Z_t& = \...
0
votes
0answers
24 views

Converting t-year probability to 1-year probability

I have a data set of peak-over-threshold values which I have fitted a generalised Pareto distribution to. From this, I wish to determine values corresponding to certain return periods in years. I have ...
1
vote
1answer
14 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 ...
-2
votes
1answer
44 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 $[0, 1]$?
1
vote
1answer
59 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 ...
1
vote
2answers
40 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 ...
1
vote
2answers
106 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 ...
1
vote
1answer
37 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 ...
0
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0answers
15 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 ...
0
votes
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,...
0
votes
2answers
54 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}(\...
3
votes
2answers
119 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 ...
0
votes
0answers
25 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'''(...
1
vote
1answer
274 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 ...
2
votes
0answers
24 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}...
1
vote
1answer
55 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-...
0
votes
1answer
340 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 ...
1
vote
1answer
679 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 ...
1
vote
2answers
75 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 $(0, 1]$ ...
0
votes
1answer
55 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 ...
2
votes
1answer
33 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
32 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 $f(x,y)$ ...
2
votes
2answers
72 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
87 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.