# 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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### 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 $x$, ...
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1 vote
263 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 ...
97 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, ...
194 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 ...
92 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 ...
39 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 vote
125 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.
274 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 ...
207 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 vote
### 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 ...
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 \$...