Gọi \(a< b\) là hai số thực bất kì
\(y\left(b\right)-y\left(a\right)=b^3-a^3-\left(b^2-a^2\right)+b-a\)
\(=\left(b-a\right)\left(b^2+ab+a^2\right)-\left(b-a\right)\left(b+a\right)+b-a\)
\(=\left(b-a\right)\left(b^2+ab+a^2-a-b+1\right)\)
\(=\left(b-a\right)\left[\left(b-\frac{1}{2}\right)^2+\left(a-\frac{1}{2}\right)^2+\frac{1}{2}\right]>0\) \(\forall b>a\)
\(\Rightarrow y\left(b\right)>y\left(a\right)\) \(\forall b>a\Rightarrow\) hàm số đồng biến trên R