Question

718.Poate sa gaseasca cineva solutiile acestei ecuatii?

Answer 1

$\displaystyle Ecuatia~se~mai~scrie~ \left( \frac{3}{16} \right)^x+ \left( \frac{1}{4} \right)^x=1. \\ \\ Membrul~stang~este~o~functie~strict~descrescatoare.~Prin~urmare \\ \\ ecuatia~admite~cel~mult~o~solutie. \\ \\ Daca~luam~f:\mathbb{R} \to \mathbb{R},~f(x)= \left( \frac{3}{16} \right)^x+ \left( \frac{1}{4} \right)^x,~observam~ca \\ \\ f(0)=2\ \textgreater \ 1,~iar~f(1)=\frac{7}{16}\ \textless \ 1. \\ \\ Deci~exista~c \in(0,1)~astfel~incat~f(c)=1.$

$\displaystyle Iar~conform~observatiei~din~randurile~2~si~3,~rezulta~ca~aceasta \\ \\ solutie~este~unica. \\ \\ Deci~raspunsul~este~B.$