# 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 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 ...
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### 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. ...
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### 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} |...
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### 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'...
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### 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 ...
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### 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}...
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### 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-...
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### 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 ...
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### 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 ...
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### 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 $(0, 1]$ ...
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### 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 ...
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### 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 ...
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### 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 $f(x,y)$ ...
### 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 $a>0$.