Question

cum se rezolva aceasta problema ?

Answer 1

$\displaystyle Avem~f'(x)+f^2(x)+1 \ge 0.~Impartim~cu~f^2(x)+1>0~si~obtinem \\ \\ \frac{f'(x)}{f^2(x)+1}+1 \ge 0 \Leftrightarrow (\arctan f(x)+x)' \ge 0.~\forall~x \in (a,b).\\ \\ Fie~g:(a,b) \to \mathbb{R},~g(x)=\arctan f(x)+x. \\ \\ Din~g'(x) \ge 0~\forall~x \in(a,b)~rezulta~ca~g~este~crescatoare. \\ \\ Deci~imaginea~functiei~g~va~fi~intervalul~(u,v),~unde \\ \\ u=\lim_{x \searrow a} g(x),~iar~v= \lim_{x \nearrow b} g(x).$

$\displaystyle Evident~trebuie~sa~avem~u<v. \\ \\ u= \arctan (+\infty)+a= \frac{\pi}{2}+a.\\ \\ v= \arctan(- \infty)+b= -\frac{\pi}{2}+b.\\ \\ u<v \Leftrightarrow \frac{\pi}{2}+a< -\frac{\pi}{2}+b \Leftrightarrow b-a> \pi \Leftrightarrow b-a \in (\pi,+\infty).~\\ \\ Raspuns:~c.$