TH1: Lấy \(x_1;x_2\in R\) sao cho \(0< x_1< x_2\)
\(\dfrac{f\left(x_1\right)-f\left(x_2\right)}{x_1-x_2}=\dfrac{a\cdot\left(x_1^2-x_2^2\right)}{x_1-x_2}=a\cdot\left(x_1+x_2\right)\)>0 vì \(x_1+x_2>0;a>0\)
=>Hàm số y=f(x)=ax2 đồng biến khi x>0 nếu a>0
TH2: Lấy \(x_1;x_2\in R^+;0< x_1< x_2\)
\(\dfrac{f\left(x_1\right)-f\left(x_2\right)}{x_1-x_2}=\dfrac{a\cdot\left(x_1^2-x_2^2\right)}{x_1-x_2}=\dfrac{a\left(x_1-x_2\right)\left(x_1+x_2\right)}{x_1-x_2}\)
\(=a\left(x_1+x_2\right)< 0\)(vì x1+x2>0 và a<0)
=>Hàm số nghịch biến khi x>0
TH3: Lấy \(x_1;x_2\in R^-\) sao cho \(x_1< x_2< 0\)
\(\dfrac{f\left(x_1\right)-f\left(x_2\right)}{x_1-x_2}=\dfrac{a\left(x_1^2-x_2^2\right)}{x_1-x_2}=\dfrac{a\left(x_1+x_2\right)\left(x_1-x_2\right)}{x_1-x_2}\)
\(=a\left(x_1+x_2\right)>0\) vì a<0 và x1+x2<0
=>Hàm số đồng biến khi x<0
