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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0answers
26 views

Maximum distribution of the Ornstein-Uhlenbeck process?

As the title says, I am searching for literature about the maximum distribution of the (stationary) Ornstein-Uhlenbeck process in finite time intervals [0,t]. Can anybody help me here? Many thanks ...
2
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0answers
34 views

Metric Spaces (Extreme Value Theorem)

I been reading Royden and Fitzpatrick's Real Analysis and in chapter 9 - 4th edition ($*$) they prove the extreme value theorem for metric spaces. First I'll transcribe the theorem and then I will ...
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1answer
34 views

Why is this successive bisection proof that proves the boundedness theorem for continuous functions correct?

I am reading Theorem 3.11 from the book "Apostol calculus Vol 1". Let $f$ be continuous on a closed internval $[a,b]$. Then $f$ is bounded on $[a,b]$. That is, there is a number $C\ge 0$ ...
0
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0answers
24 views

Find global maximum and minimum with EVT

Find the global maximum and minimum with EVT of the function $x^2$ on the (0;8] interval. I know that the maximum would be 64 and it does not have minimum just by knowing the function, but let’s say ...
2
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0answers
34 views

Show a strict global maximum for a multivariable function

I have to show that $(1,1)$ a strict global maximum for $(x,y)\rightarrow xye^{-x-y}$ with $x > 0$ and $y > 0$. First I calculated the derivatives: $$\frac{∂f}{∂x}=y(x-1)(-e^{-x-y}) $$$$\frac{∂f}...
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0answers
34 views

Uniform continuity and maximum and minimum with a continuous function in $\mathbb{R}^n$.

Problem: Let $f: \mathbb{R}^n \to \mathbb{R}$ a continuous function, so that for every $\epsilon >0$, there exits a compact set $K \subset \mathbb{R}^n$ with $|f(x)|<\epsilon$ for all $x \in \...
1
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1answer
47 views

How to prove that a Local minimum is Absolute minimum in $R^3$

Just trying to solve this question: $f(x,y,z) = x^2 + y^2 +3z^2 -xy +2xz+ yz$. Found the only critical point of the function and explain why she is an absolute minimum. We learn at class how to found ...
0
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0answers
27 views

How this limit prove the Frechet Distribution has Heavy Tails?

First, we start from the GEV-distribution function: Theorem 1.1.3 (Fisher and Tippett (1928), Gnedenko (1943)) The class of extreme value fistributions is $G_\gamma(ax+b)$ with $a>0$, $b$ real, ...
0
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0answers
12 views

Extrema of function defined on a circle not located at origo

Find the global maxima and minima of the function $$f(x,y) = \sqrt{(x-2)^2 + y-1^2} + 4x - x^2 $$ on the circle $$(x-2)^2 + y-1^2 \leq 1$$. I found $f_x$ and $f_y$ and used them to find $x=2, \sqrt{(x-...
0
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1answer
35 views

Maximum and minimum values over a triangle

I was triying to find maximum and mnimum values of $f(x,y)=xy(1-x-y)$ over the triangle with vertices $(0,0)$, $(0,1)$ and $(1,0)$. I have seen that vertices give minimum for this function. When I use ...
0
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2answers
30 views

Decompose a positive number into the sum of 5 numbers in such a way that the sum of its inverse is maximal. [closed]

I have to decompose a positive number into the sum of 5 numbers in such a way that the sum of its inverse is maximal. I understand that is optimization, but I don't know how to do it with five ...
1
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0answers
26 views

Expectation of maximum among N independent but not identical distributed guassian random variables

Suppose here are n independent gaussian random variables, $X_1,X_2\dots X_n$, such that $X_i\sim \mathcal{N}(\mu,\sigma_i^2)$, let $X^*=max(X_1,X_2\dots X_n)$. If $\sigma_1=\sigma_2=\dots\sigma_n$, I ...
1
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6answers
103 views

maximum of $x^2-3xy-2y^2$ subjects to $x^2+xy+y^2=1$ without Lagrange multiplier

What is the maximum of $x^2-3xy-2y^2$ subjects to $x^2+xy+y^2=1$? I wonder there is a precalculus method, without using the Lagrange multiplier.
3
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3answers
129 views

Global Maximum of $f(x,y)$ on a set $M$.

I want to investigate the function $f(x,y) = 4x^2 + 9y - \frac{1}{3}y^3$ on the set $M:= \{(x,y) \in \mathbb{R}^2 : y \geq |x|\}$. In particular, I want to a) proof the existence of a global maximum ...
2
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1answer
79 views

Find the extrema of $f(x,y) = x^3\cdot y^3$ in $\mathbb{R}^2$

I am asked to find the extrema of $$f(x,y) = x^3\cdot y^3$$ in $\mathbb{R}^2$ However, using the Hessian criteria, I get that the determinant of the Hessian matrix is zero for the two possible ...
1
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0answers
22 views

Ratio of conditional expectations $\frac{E[X^{\epsilon-1}|X\geq rY)}{E[Y^{\epsilon-1}|X\leq rY)}$ if $X$ and $Y$ are Fréchet

Let $X$, $Y$ be two independent continuous random variables with support on $\mathbb{R}_+$. Specifically, suppose that $X$ and $Y$ are distributed Fréchet with identical location parameter $0$ and ...
-1
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1answer
49 views

If 𝑠 is the supremum of 𝑋, prove that 𝑓(𝑎) ≤ 𝑓(𝑠) [closed]

Let $𝑓(𝑥)$ be a continuous function on a closed interval $[𝑎,𝑏]$ and let $𝑋$ be the set consisting of all $𝑎 \leq 𝑥 \leq 𝑏$ for which $𝑓(𝑎) \leq 𝑓(𝑥)$. If $𝑠$ is the supremum of $𝑋$, ...
1
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0answers
37 views

What is the extreme boundary of A in l-1?

We know that $ A=\{ f \in \ell_1 (N): f(x) \geq 0 , \sum_{x=1}^ \infty f(x)\leq 1, \sum_{x=1}^ \infty \frac{ (-1)^x f(x)}{x}=0\} $ is weak* compact and convex. Why the extreme boundary of $ A$ is $...
0
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1answer
34 views

Prove that for a given $r>0$ there exists $\theta >0$ so that for all $a \leq x \leq b$, $\theta |f(x)|\leq r$

Let $f(x)$ be a given continuous function on a closed interval $[a,b]$. Prove that for a given $r>0$ there exists $\theta > 0$ so that for all $a \leq x \leq b$, $\theta |f(x)| \leq r$. Hello, ...
0
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3answers
49 views

Continuity with $f(x)=f(x+1)$ $\forall x\in \mathbb{R}$. Is there $x \in \mathbb{R}$ with $f(x)=f(x+\sqrt{2})$?

Let $f: \mathbb{R} \to \mathbb{R}$ be continous with $f(x)=f(x+1)$ $\forall x\in \mathbb{R}$. Show there exists $x \in \mathbb{R}$ with $f(x)=f(x+\sqrt{2})$. My question is, do we even need $f(x)=f(x+...
4
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1answer
67 views

Find the extrema points of $ f\left(x\right)=\prod_{i=1}^{n}x_{i} $

Let $ f:\mathbb{R}^n \to \mathbb{R} $ given by $ f\left(x\right)=\prod_{i=1}^{n}x_{i} $ I have to find the extrema points of $ f $ under the following constraint: $ S=\left\{ \left(x_{1},...,x_{n}\...
0
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1answer
19 views

Why does the image of an interval being unbounded imply there exists a sequence with the image of the sequence greater than n?

I am trying to understand a specific step in the proof of the Extreme Value Theorem. The step is showing that the image of a continuous function over a compact interval, $f:[a,b] \rightarrow \mathbb{R}...
2
votes
1answer
39 views

On the number of extremum

Let $f$ be infinitely differentiable on $[-1,1]$, $f^{(n)}(-1)=f^{(n)}(1)=0,n=0,1,2,\cdots,$ and $f>0$ in $(-1,1)$. Prove that there is a positive integer $k$ such that $\dfrac{f(x)}{(1-x^2)^k}$ ...
0
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0answers
13 views

Reference Request: Multivariate Version of the EVT

I'm reading about extreme value theory and everythin I've seen is univariate. My question is, is there a multivariate generalization/extension of the extreme value theorem and of generalized extreme ...
0
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1answer
32 views

Extreme value theorem and half-closed intervals

I undestand that the EVT states that if we have an interval like $[a, b]$ then if there is a continuous function $f$ defined over the interval then there are such $x_1, x_2\in [a, b]$ that $f(x_1) = \...
0
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1answer
62 views

Extreme value theorem for functions defined on weakly compact sets.

So I start this by saying that I know very littler about topology and the related topics in this question, so this question may be trivial/ill-possed, etc. I've read in many places that it is a well ...
0
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1answer
81 views

Extreme values of a multivariable function lying on a helix

Suppose I've a function $f(x,y,z)$ and a helix. There are parameterised points on the helix (using the parameter $t$) such that- $\:\:\:\:\:\:\:\:\:$ $x=cos(t)$ , $y=sin(t)$ , $z=t$ Now, the ...
1
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1answer
68 views

Converse to extreme value theorem?

It's well known that continuous functions achieve minima on compact sets. An even stronger result is that lower semicontinuous functions achieve minima on compact sets. Question. If an extended-real ...
0
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0answers
21 views

Point of minimum volume of a three-dimensional region limited by a tanget plane and the graph of a convex function

My problem: Let $f:\mathbb{R}^2\to\mathbb{R}$ be a convex and differentiable function, let $P=(x_0,y_0)$ be a point in the unit closed disk $D(0,1)$ and let $z=g_{(x_0,y_0)}(x,y)$ be the tangent plane ...
0
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0answers
32 views

Find the extreme values of $f(x,y,z) = x+2y+3z$ on the circle C, $C = \{(x,y,z) \in R^3 | x^2 +y^2 = 1,z = 0\}$

Find the extreme values of $f(x,y,z) = x+2y+3z$ on the circle C in the $(x,y)$-plane of radius 1 centered at the origin; that is, $C = \{(x,y,z) \in R^3 | x^2 +y^2 = 1,z = 0\}$ I know I am supposed to ...
1
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2answers
42 views

If f: [2,5] $\rightarrow [4,13]$ is a continuous function prove that there is $c\in [2,5]$ such that $f(c) = 3c-2$

If $f: [2,5] \rightarrow [4,13]$ is a continuous function prove that there is $c\in [2,5]$ such that $f(c)= 3c-2$. I understand that the extreme value theorem is important in this question and that at ...
0
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0answers
20 views

Proof global minimum in two dimensions WITHOUT hessian but based on limit behavior

Consider a function $\Phi \colon D \to \mathbb{R}$, $D = \{(x,y) \in (0,\infty)^2 \mid x < y\}$, $\Phi$ differentiable on $D$. It is known, that $(x^\ast,y^\ast)$ is the only critical point in $D$ ...
0
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1answer
273 views

Extreme Value Theorem

For any function, given an interval with closed/open brackets, does a function have to have a min and max for the Extreme Value Theorem to apply? For example, does the extreme value theorem apply for ...
0
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1answer
32 views

Application of the Extreme Value Theorum

I understand what the Theorem means and how to apply it effectively. For example, the theorem holds true at intervals of $f(x)= \sin(x)$ at $[-2,2]$. But one thing is confusing me is if the maxima AND ...
0
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2answers
50 views

Show that a function from a closed and non-bounded set $A$ to $[0,\infty)$ has max given $\lim_{x\to \infty} f(x)=0$

If we define a continuous non-constant function $f:A\to [0,\infty)$ where $A\subset \mathbb{R^n}$ and is closed but not bounded, and if we also have that $\lim_{x\to \infty} f(x)=0$ then can we prove ...
3
votes
1answer
54 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
votes
1answer
88 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
vote
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
21 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
33 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
votes
6answers
181 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
317 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
votes
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
votes
1answer
38 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
votes
2answers
388 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 $A⊂\mathbb{R}^2$,...
0
votes
1answer
33 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
vote
1answer
63 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
18 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
27 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
vote
1answer
58 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'...