Show life that you have a thousand reasons to smile
© Copyright 2024 DOKU.TIPS - All rights reserved.
![\displaystyle Daca~notam~t= \sqrt[3]{x^2+8},~ecuatia~devine~ \\ \\ mt^2+2(m-1)t+m-1=0.~(*) \\ \\ Sa~observam~ca~t~ia~orice~valoare \ge 2.~Deci~o~conditie~necesara~si \\ \\ si~suficienta~pentru~ca~toate~radacinile~sa~fie~reale~ar~fi \\ \\ t_1 \ge 2 ~si~ t_2 \ge 2. \\ \\ Sa~mai~observam~ca~m\ \textgreater \ 0.~(In~caz~contrar~am~avea~mt^2\ \textless \ 0~; \\ \\ 2(m-1)t\ \textless \ 0~si~m-1\ \textless \ 0)~deci~egalitatea~(*)~ar~fi~imposibila). \\ \\ Cea~mai~rapida~cale~prin~care~am~demonstrat~ca~m \le \frac{5}{9}~a~fost \\ \\ urmatoarea:](https://tex.z-dn.net/?f=%5Cdisplaystyle+Daca%7Enotam%7Et%3D+%5Csqrt%5B3%5D%7Bx%5E2%2B8%7D%2C%7Eecuatia%7Edevine%7E+%5C%5C++%5C%5C+mt%5E2%2B2%28m-1%29t%2Bm-1%3D0.%7E%28%2A%29+%5C%5C++%5C%5C+Sa%7Eobservam%7Eca%7Et%7Eia%7Eorice%7Evaloare+%5Cge+2.%7EDeci%7Eo%7Econditie%7Enecesara%7Esi+%5C%5C++%5C%5C+si%7Esuficienta%7Epentru%7Eca%7Etoate%7Eradacinile%7Esa%7Efie%7Ereale%7Ear%7Efi+%5C%5C++%5C%5C+t_1+%5Cge+2+%7Esi%7E+t_2+%5Cge+2.+%5C%5C++%5C%5C+Sa%7Emai%7Eobservam%7Eca%7Em%5C+%5Ctextgreater+%5C+0.%7E%28In%7Ecaz%7Econtrar%7Eam%7Eavea%7Emt%5E2%5C+%5Ctextless+%5C+0%7E%3B+%5C%5C++%5C%5C+2%28m-1%29t%5C+%5Ctextless+%5C+0%7Esi%7Em-1%5C+%5Ctextless+%5C+0%29%7Edeci%7Eegalitatea%7E%28%2A%29%7Ear%7Efi%7Eimposibila%29.+%5C%5C++%5C%5C+Cea%7Emai%7Erapida%7Ecale%7Eprin%7Ecare%7Eam%7Edemonstrat%7Eca%7Em+%5Cle+%5Cfrac%7B5%7D%7B9%7D%7Ea%7Efost+%5C%5C++%5C%5C+urmatoarea%3A+ "\displaystyle Daca~notam~t= \sqrt[3]{x^2+8},~ecuatia~devine~ \\ \\ mt^2+2(m-1)t+m-1=0.~(*) \\ \\ Sa~observam~ca~t~ia~orice~valoare \ge 2.~Deci~o~conditie~necesara~si \\ \\ si~suficienta~pentru~ca~toate~radacinile~sa~fie~reale~ar~fi \\ \\ t_1 \ge 2 ~si~ t_2 \ge 2. \\ \\ Sa~mai~observam~ca~m\ \textgreater \ 0.~(In~caz~contrar~am~avea~mt^2\ \textless \ 0~; \\ \\ 2(m-1)t\ \textless \ 0~si~m-1\ \textless \ 0)~deci~egalitatea~(*)~ar~fi~imposibila). \\ \\ Cea~mai~rapida~cale~prin~care~am~demonstrat~ca~m \le \frac{5}{9}~a~fost \\ \\ urmatoarea:")
