b, Lấy \(x_1;x_2\in\left(-\infty;2\right)\left(x_1\ne x_2\right)\)
\(\Rightarrow y_1=\frac{3}{2-x_1};y_2=\frac{3}{2-x_2}\)
\(\Rightarrow y_1-y_2=\frac{3}{2-x_1}-\frac{3}{2-x_2}=\frac{3\left(2-x_2-2+x_1\right)}{\left(2-x_1\right)\left(2-x_2\right)}=\frac{3\left(x_1-x_2\right)}{\left(2-x_1\right)\left(2-x_2\right)}\)
\(\Rightarrow\frac{y_1-y_2}{x_1-x_2}=\frac{3}{\left(2-x_1\right)\left(2-x_2\right)}\)
Do \(x_1;x_2\in\left(-\infty;2\right)\Rightarrow\left(2-x_1\right)\left(2-x_2\right)>0\)
\(\Rightarrow I=\frac{y_1-y_2}{x_1-x_2}=\frac{3}{\left(2-x_1\right)\left(2-x_2\right)}>0\)
\(\Rightarrow\) Hàm số đồng biến trên \(\left(-\infty;2\right)\)
Lấy \(x_1;x_2\in\left(2;+\infty\right)\left(x_1\ne x_2\right)\)
\(\Rightarrow y_1=\frac{3}{2-x_1};y_2=\frac{3}{2-x_2}\)
\(\Rightarrow y_1-y_2=\frac{3}{2-x_1}-\frac{3}{2-x_2}=\frac{3\left(2-x_2-2+x_1\right)}{\left(2-x_1\right)\left(2-x_2\right)}=\frac{3\left(x_1-x_2\right)}{\left(2-x_1\right)\left(2-x_2\right)}\)
\(\Rightarrow\frac{y_1-y_2}{x_1-x_2}=\frac{3}{\left(2-x_1\right)\left(2-x_2\right)}\)
Do \(x_1;x_2\in\left(-\infty;2\right)\Rightarrow\left(2-x_1\right)\left(2-x_2\right)>0\)
\(\Rightarrow I=\frac{y_1-y_2}{x_1-x_2}=\frac{3}{\left(2-x_1\right)\left(2-x_2\right)}>0\)
\(\Rightarrow\) Hàm số đồng biến trên \(\left(2;+\infty\right)\)
a, Lấy \(x_1;x_2\in\left(-\infty;-1\right)\left(x_1\ne x_2\right)\)
\(\Rightarrow y_1=\frac{4}{x_1+1};y_2=\frac{4}{x_2+1}\)
\(\Rightarrow y_1-y_2=\frac{4}{x_1+1}-\frac{4}{x_2+1}=\frac{4\left(x_2+1-x_1-1\right)}{\left(x_1+1\right)\left(x_2+1\right)}=-\frac{4\left(x_1-x_2\right)}{\left(x_1+1\right)\left(x_2+1\right)}\)
\(\Rightarrow\frac{y_1-y_2}{x_1-x_2}=-\frac{4}{\left(x_1+1\right)\left(x_2+1\right)}\)
Do \(x_1;x_2\in\left(-\infty;-1\right)\Rightarrow\left(x_1+1\right)\left(x_2+1\right)>0\)
\(\Rightarrow I=\frac{y_1-y_2}{x_1-x_2}=-\frac{4}{\left(x_1+1\right)\left(x_2+1\right)}< 0\)
\(\Rightarrow\) Hàm số nghịch biến trên \(\left(-\infty;-1\right)\)
Lấy \(x_1;x_2\in\left(-1;+\infty\right)\left(x_1\ne x_2\right)\)
\(\Rightarrow y_1=\frac{4}{x_1+1};y_2=\frac{4}{x_2+1}\)
\(\Rightarrow y_1-y_2=\frac{4}{x_1+1}-\frac{4}{x_2+1}=\frac{4\left(x_2+1-x_1-1\right)}{\left(x_1+1\right)\left(x_2+1\right)}=-\frac{4\left(x_1-x_2\right)}{\left(x_1+1\right)\left(x_2+1\right)}\)
\(\Rightarrow\frac{y_1-y_2}{x_1-x_2}=-\frac{4}{\left(x_1+1\right)\left(x_2+1\right)}\)
Do \(x_1;x_2\in\left(-1;+\infty\right)\Rightarrow\left(x_1+1\right)\left(x_2+1\right)>0\)
\(\Rightarrow I=\frac{y_1-y_2}{x_1-x_2}=-\frac{4}{\left(x_1+1\right)\left(x_2+1\right)}< 0\)
\(\Rightarrow\) Hàm số nghịch biến trên \(\left(-\infty;-1\right)\)
