# 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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### 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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### 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 ...
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 $C\ge 0$ ...
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 ...
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### 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 ...
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 ...
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### 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 ...
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 ...