# 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 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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### 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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### 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 ...
• 389
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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 ...
• 710
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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 ...
1 vote
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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 $C\ge 0$ ...
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 ...