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.
157
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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 $x$, ...
- 33
2
votes
1
answer
69
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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}((...
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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 ...
- 35
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1
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52
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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 ...
- 13
3
votes
1
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35
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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 ...
- 749
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19
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Fitting a general extreme value distribution on transformed data
I have a data set $X_1, ..., X_n$ and want to fit a general value distribution to this data set using R.
However, when trying to do so I get an overflow. So, my idea was to instead fit a general ...
- 75
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118
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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.)...
- 2,862
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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
$$P = \left |3 i (...
- 193
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1
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48
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Is the point we get from using lagrange considered a critical point?
Find the critical points of $f (x, y) = 2xy$ when restricted to $$ x^2 + 2y^2 = 6$$
I just wanna make sure I understand one thing, in the answer they provided the values they got after doing lagrange. ...
- 1,015
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19
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Can we apply the extreme value theorem in this true-false question?
Problem. Is it true that $h(x) = f(g(x))$ has global maximum or global minimum if $g$ is continuous in $[a,b]$, $f$ is continuous in $[c,d]$ with the image of $f$ contained in $[a,b]$?
Attempt. I ...
- 667
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1
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22
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Given a two-variable function, determine that a given set represents the set of critical points for the function
The function in question would be $f: \mathbb{R}^2 \to \mathbb{R}, (x,y)
\mapsto x^2y^2 +\sin{y} $. Show that the given set $$ \left \{ \left( 0, \dfrac{\pi}{2} + k\pi \right) : k \in \mathbb{Z} \...
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40
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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 ...
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52
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How to evaluate the conditional mean of Frechet distribution?
I am trying to get the derivation of the following conditional mean of a Frechet distributed variable.
$r^{k} A^{k}(\omega)$ is i.i.d. Frechet distributed with shape parameter $\theta>1$ and scale ...
- 1
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0
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64
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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 ...
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353
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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 ...
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2
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175
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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 ...
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41
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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 ...
2
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2
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146
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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:
$$
C^2 = 18 + 8x - 2x^2 + 2\sqrt{5x - x^2}...
- 51
1
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0
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54
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The maximum of the sum of fixed-length substrings in a Bernoulli process
Suppose a biased coin (probability of head being $p$) was flipped $n$ times. For a fixed positive integer $k$ ($k<n$), there exists $(n-k+1)$ consecutive subsequences. I am interested in the ...
- 11
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206
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What's the difference between Boundedness Theorem and Extreme Value Theorem?
I know that the Boundedness Theorem states that if a function is continuous on a closed interval then it is bounded on that interval, but doesn't it mean that it also attains the maximum and minimum ...
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answers
67
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Explosive Cox process
Let $X_0,\dots,X_n,\dots$ be a sequence of independent variables and let each $X_k$ be distributed according to an exponential distribution of parameter $\lambda a^{-k}$ (with $a<1$):
$$X_k \sim \...
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53
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(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 ...
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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
...
- 321
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29
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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 $...
- 111
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45
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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 ...
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29
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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\...
- 449
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51
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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}(\...
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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$ ...
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143
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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{...
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506
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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 ...
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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 $(...
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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 \...
- 324
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0
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32
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Extreme-value problem of multi-variable function
I am reading a paper in which they solve the following problem;
start
$$
\begin{aligned}
\mathcal{W}_{\alpha_{1}, \beta_{1}, \theta} &=\frac{1}{2}\left[\cos \beta_{1} \cos \alpha_{1}+\cos \beta_{...
- 324
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2
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93
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Use the Mean Value Theorem to prove an Inequality. [closed]
Use the Mean Value Theorem to show that $$\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}}$$ I ...
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214
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Maximum distribution of the Ornstein-Uhlenbeck process?
As the title says, I am searching for literature about the maximum distribution of the (stationary) Ornstein-Uhlenbeck process in finite time intervals [0,t]. Can anybody help me here?
Many thanks
...
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310
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Metric Spaces (Extreme Value Theorem)
I been reading Royden and Fitzpatrick's Real Analysis and in chapter 9 - 4th edition ($*$) they prove the extreme value theorem for metric spaces.
First I'll transcribe the theorem and then I will ...
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109
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Why is this successive bisection proof that proves the boundedness theorem for continuous functions correct?
I am reading Theorem 3.11 from the book "Apostol calculus Vol 1". Let $f$ be continuous on a closed internval $[a,b]$. Then $f$ is bounded on $[a,b]$. That is, there is a number $C\ge 0$ ...
- 151
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129
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Find global maximum and minimum with EVT
Find the global maximum and minimum with EVT of the function $x^2$ on the (0;8] interval. I know that the maximum would be 64 and it does not have minimum just by knowing the function, but let’s say ...
- 57
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97
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Show a strict global maximum for a multivariable function
I have to show that $(1,1)$ a strict global maximum for $(x,y)\rightarrow xye^{-x-y}$ with $x > 0$ and $y > 0$.
First I calculated the derivatives: $$\frac{∂f}{∂x}=y(x-1)(-e^{-x-y})
$$$$\frac{∂f}...
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answers
152
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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 \...
- 157
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1
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263
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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 ...
- 77
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0
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97
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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, ...
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1
answer
194
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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 ...
- 1,291
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2
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92
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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 ...
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2
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0
answers
39
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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 ...
- 31
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6
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125
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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.
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3
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274
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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 ...
- 984
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1
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207
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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 ...
- 23
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vote
0
answers
131
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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 ...
- 11
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0
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44
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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 $...
- 127