Question Number 36927 by maxmathsup by imad last updated on 07/Jun/18
![f is a real function derivable on [0,1] /f(0)=0 and f(1)=1 prove that ∀n∈N ∃ (x_i )_(1≤i≤n) seqence of reals with x_i ≠x_j if i≠j and Σ_(k=1) ^n f^′ (x_k )=n.](https://www.tinkutara.com/question/Q36927.png)
$${f}\:{is}\:{a}\:{real}\:{function}\:{derivable}\:{on}\:\left[\mathrm{0},\mathrm{1}\right]\:/{f}\left(\mathrm{0}\right)=\mathrm{0}\:{and}\:{f}\left(\mathrm{1}\right)=\mathrm{1} \\ $$$${prove}\:{that}\:\forall{n}\in{N}\:\:\exists\:\:\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}} \:{seqence}\:{of}\:{reals}\:{with}\:{x}_{{i}} \neq{x}_{{j}} \:{if}\:{i}\neq{j} \\ $$$${and}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{f}^{'} \left({x}_{{k}} \right)={n}. \\ $$