Let $f$ be any function asymptotic at zero. Prove/Disprove:

If $f$ is differentiable everywhere in its domain, then $\lim_{x\to \infty}f\;'(x)$ must be equal to zero?

If $f$ is differentiable everywhere in its domain, then $\lim_{x\to \infty}f\;'(x)$ must be equal to zero?

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Consider a game with three vertical sticks and 5 circles, where
each circle has a hole in the middle. The circles arranged in
one stick as shown in the picture:
The task is to move all the circles to a nearby stick,
circles can be move from one stick to another, but a
circle cannot be put over a smaller one. What are the
minimum number of steps needed to complete this task?

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The sequence of Fibonacci can be found everywhere in nature. First mentioned by Leonardo Fibonacci who was a medieval mathematician that was trying to solve problem involving the growth of a population of rabbits. It became one of the most famous sequence in all of mathematics. Here is a mathematical description of some of its fundamental properties...

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