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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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How to show $ \lim_{n\rightarrow \infty}\frac{E[\max_{1,2...,n} X_i]}{\sqrt{2\ln n}} = 1$ where $(X_i)$ is a sequence of iid gaussian rvs? [duplicate]

Let $(X_i)$ be a sequence of iid $N(0,1)$ random variables. I would like to show that $$ \lim_{n\rightarrow \infty}\frac{E[\max_{1,2...,n} X_i]}{\sqrt{2\ln n}} = 1. $$ My first try was to find $a_n, ...
ProbabilityLearner's user avatar
1 vote
2 answers
105 views

Prove $(x_1 +x_2 +...+x_i +...+x_n)^2 \ge n(x_1x_2 +x_2x_3 +...+x_ix_{i+1} +...+x_nx_1)$

Determine all positive integers $n \ge 2$ such that for all POSITIVE real number $x_1, x_2,..., x_n$ the following inequality holds: $$(x_1 +x_2 +...+x_i +...+x_n)^2 \ge n(x_1x_2 +x_2x_3 +...+x_ix_{i+...
Kokos's user avatar
  • 418
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0 answers
19 views

Number of minima and maxima of a scalar function in $\mathbb R^3$

Given a continuous (if need be $C^\infty$) function $\Phi(\boldsymbol{x})$, mapping $\mathbb{R}^3\to\mathbb{R}$, that has $n$ minima and obtains $\Phi\to0$ for $|\boldsymbol{x}|\to\infty$, is there a ...
Walter's user avatar
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1 vote
2 answers
89 views

Extreme Value Theorem on Logistic Distribution

Let $𝑋_1,𝑋_2, … , 𝑋_𝑛$ be a random sample of size 𝑛 from a distribution with cumulative distribution function $F(𝑥) = \frac{1}{1 + \exp(-cx)}$, $−\infty < 𝑥 < \infty$, where $𝑐 > 0$. ...
john22445's user avatar
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0 answers
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Find the absolute and relative extrema of $f(x,y,)=8xy+y$, over the region $0≤y≤15-x$, $0≤x≤5$

Firstly, I find the critical points: $f_x=8y=0$; $f_y=8x+1=0$ From here, $y=0$ and $x= -{{1} \over 8}$ and I find the point $P(0, -{{1} \over 8})$ which, does not satisfy the condition under $x$, ...
Vile's user avatar
  • 53
2 votes
1 answer
71 views

prove that $f^{-1}((0,0))$ is infinite

Let $A = \{(x,y) \in \mathbb{R}^2 : x = 0 \text{ or } y = 0\}$ equipped with the subspace topology from $\mathbb{R}^2$. Let $f:\mathbb{R}\to A$ be a surjective and continuous map. Prove that $f^{-1}((...
Alfred's user avatar
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3 votes
1 answer
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Find $a_n,b_n$ so that $b_n(X_n-a_n)$ converges to a non-degenerate limit

I have the following problem: Let $X_n$ be the maximum of a random sample $Y_1,...,Y_n$ from the density $f(x)=2(1-x), x\in [0,1]$. Find constants $a_n,b_n$ so that $b_n(X_n-a_n)$ converges in ...
user_42's user avatar
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1 answer
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How to determine if derivative changes sign when it's undefined at a point?

I had to find the absolute extrema of $f(x) = 3\sqrt[3]{x^2}-2x$ in the range $[-1;2]$. So first I found that $f'(x) = 2x^{-\frac{1}{3}}-2$. Then I found it to be equal to 0 at the point 1. I then ...
Mihail Mihov's user avatar
3 votes
1 answer
51 views

Is my understanding of extreme value theorem correct?

The extreme value theorem says a continuous function achieves its maximum and minimum on a compact domain. Consider $f: (0,1)\rightarrow \mathbb{R}$ with $f(x)=x$. I impose the standard topology on ...
Ypbor's user avatar
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3 votes
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If $f''(x) + f'(x)g(x) - f(x) = f(a) = f(b) = 0$, then $f = 0$ on $[a,b]$

Let $f, g : \mathbb R \to \mathbb R$, with $f$ twice differentiable, and $f(a) = f(b) = 0, a \leq b$. Show that if $f''(x) + f'(x)g(x) - f(x) = 0$, then $f = 0$ on $[a,b]$. (Source: Spivak Calculus.)...
SRobertJames's user avatar
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4 votes
2 answers
94 views

Find greatest value of $P = \left |3 i (z_1 + z_2) + 9 - z_1z_2\right |.$

Let be two complexs numbers $z_1$, $z_2$ satisfying the conditions $|z_1| = 1$, $|z_2| = 1$, $\left| z_1 + z_2\right| =\sqrt{2}$. Find the least and the greatest value of the value $$P = \left |3 i (...
John Paul Peter's user avatar
0 votes
1 answer
66 views

Is the point we get from using lagrange considered a critical point?

Find the critical points of $f (x, y) = 2xy$ when restricted to $$ x^2 + 2y^2 = 6$$ I just wanna make sure I understand one thing, in the answer they provided the values they got after doing lagrange. ...
Need_MathHelp's user avatar
0 votes
0 answers
21 views

Can we apply the extreme value theorem in this true-false question?

Problem. Is it true that $h(x) = f(g(x))$ has global maximum or global minimum if $g$ is continuous in $[a,b]$, $f$ is continuous in $[c,d]$ with the image of $f$ contained in $[a,b]$? Attempt. I ...
user1145880's user avatar
0 votes
1 answer
22 views

Given a two-variable function, determine that a given set represents the set of critical points for the function

The function in question would be $f: \mathbb{R}^2 \to \mathbb{R}, (x,y) \mapsto x^2y^2 +\sin{y} $. Show that the given set $$ \left \{ \left( 0, \dfrac{\pi}{2} + k\pi \right) : k \in \mathbb{Z} \...
pron1ghtmare360's user avatar
0 votes
0 answers
45 views

Is there any simple way to find the extreme points to the equation $f(x)= x^2(\sin(1/x)+\cos(1/x))$when $x \in [-1,0) \cup (0,1]$ and $0$ when $x = 0$

Hey I have found the derivative to this equation: $$2x\cdot\sin\left(\frac{1}{x}\right)+2x\cdot\cos\left(\frac{1}{x}\right)+\sin\left(\frac{1}{x}\right)-\cos\left(\frac{1}{x}\right)$$ The things is ...
SliVetle's user avatar
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proof of fundamental theorem of calculus

hi i have problem understanding the proof of fundamental theorem of calculus. this is part of general proof process using extreme value. briefly this above saying that the derivate of integral is ...
KIM CHANGJUN's user avatar
0 votes
2 answers
475 views

Find the extreme values of $f(x,y)=e^{-xy}$ on the region described by $x^2+25y^2\leq 4$

I have been stuck on this question for a very long time. I have tried to use lagrange multipliers but the equation seems nearly impossible to solve as the derivative of $f$ with respect to $x$ and the ...
Kitsu Jirunabe's user avatar
1 vote
2 answers
259 views

If $f$ goes to infinity, prove it has a minimum

The question is: $f: A\to R$ is a continuous, real-valued function, where $A\subseteq\mathbb{R}^n$. If $f(x)\to\infty$ as $\|x\|\to\infty,$ show that $f$ attains a minimum. Where I’ve gotten so far ...
Arvin's user avatar
  • 25
1 vote
1 answer
59 views

How to find any absolute extrema on $\mathbb R^2$ for $y^4 - x^5$

So far I have found the critical point $(0, 0)$ and the partials of $-5x^4$ and $4y^3$. How would I proceed after finding the critical point, since I don't have a definite interval. I get confused ...
maria guallpa's user avatar
2 votes
2 answers
148 views

Find the max value of $\sqrt{5x - x^2} + \sqrt{18 + 3x - x^2}$ [closed]

I have this expression: $C = \sqrt{5x - x^2} + \sqrt{18 + 3x - x^2}$ And I need to find the max value of $C$, can anyone help me? I tried something like this: $$ C^2 = 18 + 8x - 2x^2 + 2\sqrt{5x - x^2}...
AquaPI's user avatar
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1 vote
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The maximum of the sum of fixed-length substrings in a Bernoulli process

Suppose a biased coin (probability of head being $p$) was flipped $n$ times. For a fixed positive integer $k$ ($k<n$), there exists $(n-k+1)$ consecutive subsequences. I am interested in the ...
Riki Jin's user avatar
5 votes
1 answer
273 views

What's the difference between Boundedness Theorem and Extreme Value Theorem?

I know that the Boundedness Theorem states that if a function is continuous on a closed interval then it is bounded on that interval, but doesn't it mean that it also attains the maximum and minimum ...
AdamMazur's user avatar
2 votes
0 answers
72 views

Explosive Cox process

Let $X_0,\dots,X_n,\dots$ be a sequence of independent variables and let each $X_k$ be distributed according to an exponential distribution of parameter $\lambda a^{-k}$ (with $a<1$): $$X_k \sim \...
Davius's user avatar
  • 905
0 votes
0 answers
57 views

(extremum)Let $f: \mathbb R \to \mathbb R$ be a polynomial function ...

Let $f: \mathbb R \to \mathbb R$ be a polynomial function $f = a_0 + a_1x + … + a_n x^n$. Let $a_1 = a_2 = … = a_k = 0$, ($k$ less than n) and $a_{k+1} \ne 0$. The function $f$ has an extremum at the ...
Gab's user avatar
  • 97
2 votes
1 answer
181 views

Limit of expectation of maximum of standard normal distribution

I cannot find the way to show that if $X_1,\dots,X_n$ are independent standard normal random variables, then $$ \lim_{n \rightarrow \infty} \frac {\mathbb{E} \max_{i=1,\dots,n}X_i}{\sqrt{2\log n}} = 1 ...
someeed's user avatar
  • 469
1 vote
0 answers
38 views

What does: "moments only exist for order less than $1 / \xi$" mean? (concerning the Fréchet distribution)

Could someone please help me out with this sentence. Whilst I understand the first part, I don't understand the last part in bold. "The Fréchet distribution ($\xi > 0$) has a heavy tail with $...
AlexLee's user avatar
  • 111
0 votes
2 answers
45 views

Find a cubic function given the inflection point and minimal point

I came across this task that I just couldn't figure out. The task gave me an extremum(1,1) and the inflection point(2,3), and I need to figure out the cubic function given the values. Assuming the ...
ckvywk's user avatar
  • 25
1 vote
0 answers
29 views

Higher order function's local extremum

Find ỉnteger values of $m$ such that the function: $f(x)=x^8 + (m - 2)x^5 - (m^2 - 4)x^4 + 1$ reaches local minimum at $x=0$ This is what I did: We have: $f'(x)=8x^7+5(m-2)x^4+4(4-m^2)x^3=x^3\...
Snek's user avatar
  • 449
1 vote
0 answers
58 views

Upper bound the probability that the maximum of i.i.d. r.v.'s (e.g. busy periods) exceeding a threshold

Suppose that $B_1, B_2,\ldots, B_n$ are a series of positive independent and identically distributed random variables. The moment generating function (MGF) of $B_i$'s is known, denoted as $M_{B}(\...
leeyee's user avatar
  • 317
1 vote
1 answer
535 views

An endpoint of a closed interval where the derivative is zero is considered a critical point?

Consider the function $f(x)=x-\sin(x)$. I want to find the critical points of $f(x)$ on $[0,2\pi]$. Since $f'(x)=1-\cos(x)$ then the equations $f'(x)=0$ for $x\in [0,2\pi]$ gives two solutions $x_1=0$ ...
palio's user avatar
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0 votes
0 answers
194 views

What is the the maximum and minimum of a sequence of $n$ random variables having Chi-squared distribution with $k$ degrees of freedom?

Say we have a sequence of $n$ random variables $[X_1,X_2,\ldots,X_n]$, identically distributed, having Chi-squared distribution with $k$ degrees of freedom. Then, what is the upper bound of $\underset{...
Jack2018's user avatar
0 votes
1 answer
526 views

How can I find the absolute maximum value of the function given that both $a$ and $b$ are both positive constants

I am given this function: $f(x)= x^a(1-x)^b$ and I am told that I have to find the absolute maximum value of this function within the interval of $0 \leq x \leq 1.$ Assume that both $a$ and $b$ are ...
myts999's user avatar
  • 95
1 vote
1 answer
77 views

Given $f: \mathbf{R} \to \mathbf{R}$ is continuous and $\lim_{x\to -\infty}f(x) = \infty = \lim_{x\to \infty} f(x)$, show $f$ attains its minimum.

There are plenty of questions on stack exchange similar to this but my question comes in a minute detail. My attempt: Let $a\in \mathbf{R}$ so that there are $c_1$, $c_2 \in \mathbf{R}$ such that $(...
Chris Christopherson's user avatar
3 votes
1 answer
78 views

Doubt on a method used in an article ( extreme value problem of trig function)

This might be a straightforward problem but I couldn't figure it out on my own. start $$ \begin{aligned} \mathcal{W}_{\mathbf{p}, \theta} &=\frac{1}{2}\left[\cos \alpha+\cos (\theta-\alpha)+2 \...
Dotman's user avatar
  • 326
1 vote
0 answers
32 views

Extreme-value problem of multi-variable function

I am reading a paper in which they solve the following problem; start $$ \begin{aligned} \mathcal{W}_{\alpha_{1}, \beta_{1}, \theta} &=\frac{1}{2}\left[\cos \beta_{1} \cos \alpha_{1}+\cos \beta_{...
Dotman's user avatar
  • 326
0 votes
2 answers
99 views

Use the Mean Value Theorem to prove an Inequality. [closed]

Use the Mean Value Theorem to show that $$\frac{1}{9}<\sqrt{66}-8<\frac{1}{8}$$ I really don't know how to use the mean value theorem for this as $$\frac{d}{dx}\sqrt{x}\frac{1}{2\sqrt{x}}$$ I ...
Copywright's user avatar
3 votes
0 answers
296 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 ...
Karl's user avatar
  • 710
2 votes
1 answer
431 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 ...
Giordano Ribeiro's user avatar
1 vote
1 answer
124 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 $C\ge 0$ ...
Dachuan Huang's user avatar
0 votes
0 answers
157 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 ...
GZanotto's user avatar
2 votes
0 answers
108 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}...
Arius's user avatar
  • 67
0 votes
0 answers
192 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 \...
Mark Lauer's user avatar
1 vote
1 answer
336 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 ...
yuval's user avatar
  • 79
0 votes
0 answers
110 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, ...
Jelmer's user avatar
  • 1
0 votes
1 answer
255 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 ...
user519955's user avatar
  • 1,313
0 votes
2 answers
105 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 ...
user895583's user avatar
2 votes
0 answers
43 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 ...
MissCrystal's user avatar
1 vote
6 answers
128 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.
Shara's user avatar
  • 485
3 votes
3 answers
322 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 ...
offline's user avatar
  • 974
2 votes
1 answer
236 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 ...
Gema Cabero's user avatar