Question

Ajutati-ma cu Al 79, exercitiu din culegerea de probleme pentru admitere in Universitatea Politehnica Timisoara.

Answer 1

$\displaystyle Pentru ~x\ \textless \ 0~notam~t= \left( \frac{1}{2} \right)^x\ \textgreater \ 1. \\ \\ Deci~trebuie~sa~gasim~m~real~astfel~incat~sa~avem \\ \\ t^2-mt+1\ \textgreater \ 0~ \forall~t\ \textgreater \ 1. \\ \\ ---------------------------- \\ \\ Impartim~inecuatia~prin~t\ \textgreater \ 1,~si~obtinem: \\ \\ t+ \frac{1}{t}\ \textgreater \ m~\forall~t\ \textgreater \ 1. \\ \\ Din~inegalitatea~mediilor,~avem~t+ \frac{1}{t}\ \textgreater \ 2 \sqrt{t \cdot \frac{1}{t}}=2,~inegalitatea \\ \\ fiind~stricta,~caci~aceasta~ar~avea~loc~daca~t= \frac{1}{t},~imposibil.$

$\displaystyle Din~t+ \frac{1}{t}\ \textgreater \ 2,~rezulta~ca~m \leq 2.~- \boxed{varianta~f} \\ \\ *** \\ \\ Fac~mentiunea~ca~nu~putem~avea~m\ \textgreater \ 2,~deoarece,~functia~ \\ \\ f: (1,+ \infty) \rightarrow \mathbb{R}~este~strict~crescatoare~si~bijectiva. \\ \\ Astfel,~daca~ar~exista~un~m\ \textgreater \ 2~fixat,~pentru~t~suficient~de~mic \\ \\ (t\ \textgreater \ 1)~vom~avea~t+ \frac{1}{t} \leq m.$