Với 2 số thực bất kì \(x_1;x_2\) sao cho \(x_1< x_2\) ta có:
\(f\left(x_1\right)-f\left(x_2\right)=-x_1^3+x_1^2-x_1+5-\left(-x_2^3+x_2^2-x_2+5\right)\)
\(=x_2^3-x_1^3+x_1^2-x_2^2-x_1+x_2\)
\(=\left(x_2-x_1\right)\left(x_1^2+x_2^2+x_1x_2\right)-\left(x_2-x_1\right)\left(x_1+x_2\right)+x_2-x_1\)
\(=\left(x_2-x_1\right)\left(x_1^2+x_2^2+x_1x_2-x_1-x_2+1\right)\)
\(=\left(x_2-x_1\right)\left[\left(x_1+\dfrac{x_2}{2}-\dfrac{1}{2}\right)^2+\dfrac{3}{4}x_2^2-\dfrac{1}{2}x_2+\dfrac{3}{4}\right]\)
\(=\left(x_2-x_1\right)\left[\left(x_1+\dfrac{x_2}{2}-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\left(x_2-\dfrac{1}{3}\right)^2+\dfrac{2}{3}\right]>0\)
\(\Rightarrow f\left(x_1\right)>f\left(x_2\right)\Rightarrow\) hàm nghịch biến trên R