1) Ta có:
\(P=\frac{2x}{x^2+1}\)
\(P=\frac{-x^2-1+x^2+2x+1}{x^2+1}\)
\(P=\frac{-x^2-1}{x^2+1}+\frac{x^2+2x+1}{x^2+1}\)
\(P=-1+\frac{\left(x+1\right)^2}{x^2+1}\ge-1\)
Vậy: GTNN của P là -1 khi \(x\ne1=0\Leftrightarrow x\) vô nghiệm ( vì \(x^2>0\))
2)
\(\Leftrightarrow\left\{\begin{matrix}2ab+2ac+2bc=2\\2a^2+2b^2+2c^2=2\end{matrix}\right.\)
\(\Rightarrow2a^2+2b^2+2c^2-2ab-2ac-2bc=0\)
\(\Rightarrow\left(a^2-2ab+b^2\right)+\left(a^2-2ac+c^2\right)+\left(b^2-2bc+c^2\right)=0\)
\(\Rightarrow\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\)
\(\Rightarrow\left\{\begin{matrix}a-b=0\\a-c=0\\b-c=0\end{matrix}\right.\)\(\Leftrightarrow a=b=c\)
Từ \(a^2+b^2+c^2=1\)
\(\Rightarrow3a^2=1\)
\(\Rightarrow a^2=\frac{1}{3}\)
\(\Rightarrow a=\frac{\sqrt{3}}{3}\)
\(\Rightarrow M=a+b+c=3a=3.\frac{\sqrt{3}}{3}=\sqrt{3}\)
Vậy: \(M=\sqrt{3}\)
Chúc bạn học tốt!!!
