Y by
- Let
and
be a strictly monotonous function such that
. Show that
.
- Find all convergent sequences
for which there exists a scrictly monotonous function
such that
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[list=a] [*] Let $a<b$ and $f\colon [a,b]\rightarrow\mathbb{R}$ be a strictly monotonous function such that $\int_a^b f(x) \mathrm d x=0$. Show that $f(a)\cdot f(b)<0$. [*] Find all convergent sequences $(a_n)_{n\geq 1}$ for which there exists a strictly monotonous function $f\colon\mathbb{R}\rightarrow\mathbb{R}$ such that \[\int_{a_{n-1}}^{a_n} f(x)\mathrm dx = \int_{a_n}^{a_{n+1}} f(x)\mathrm dx,\text{ for all }n\geq 2.\] [/list]
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