Question

Tìm m để phương trình \(\log_5\left(x^2+1\right)+1\ge\log_5\left(mx^2+4x+m\right)\) nghiệm đúng với mọi x thuộc R

Answer 1

\(\Leftrightarrow log_5\left(5x^2+5\right)\ge log_5\left(mx^2+4x+m\right)\) ; \(\forall x\)

\(\Leftrightarrow\left\{{}\begin{matrix}mx^2+4x+m>0\\5x^2+5\ge mx^2+4x+m\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m>\dfrac{-4x}{x^2+1}=f\left(x\right)\\m\le\dfrac{5x^2-4x+5}{x^2+1}=g\left(x\right)\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m>\max\limits_{x\in R}f\left(x\right)=2\\m\le\min\limits_{x\in R}g\left(x\right)=3\end{matrix}\right.\) 

\(\Rightarrow2< m\le3\)