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SPECIAL PROBLEMS of MATHEMATICAL ANALYSIS for COMPETITIONS

 

 

         1.[2022.06.14]

               If \;f\;:\;[a,b]\;\rightarrow \;\mathbb{R}\; is monotonic, then

\int_a^b \left | f(x)-f \left (\frac{a+b}{2} \right ) \right | dx \;\leqslant \;\int_a^b \left | f(x)-c \right | dx,

 

          whatever the number c \in \mathbb{R}.

 

          2.[2022.07.29]

                Let f\;:\;(0,a)\;\rightarrow\;\mathbb{R} \;\;(a>0) be a continuous function 

such that 0<f(x)<x for x \in (0,a). We define by recurrence 

f_1(x)=f(x),\;f_{n+1}(x)=f(f_n(x)),\;n=1,2,\dots

           If for some number m>0 there exist

\lim_{x \to 0+}\;[ ( f(x))^{-m}-x^{-m}]=p>0,

 then for any x \in(0,a) we have

\lim_{n \to \infty}\;(n\cdot p)^{m^{-1}} \cdot f_n(x) =1.

 

  3.[]

                Let




 


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