Ta có: $\text{f}\left(\text{x}\right)<\text{x}-2 \text{m}\iff \frac{1}{2}(\text{x}-\text{f}(\text{x}\left)\right)>\text{m};\forall \text{x}\in (0;2)$.

Đặt $g\left(x\right)=\frac{1}{2}[x-f(x\left)\right]\Rightarrow g^{\text{'}}\left(x\right)=\frac{1}{2}\left(1-f^{\text{'}}\left(x\right)\right)$.

Dựa vào đồ thị f'(x), ta thấy: $\forall \text{x}\in (0;2);{\text{f}}^{\text{'}}\left(\text{x}\right)<1\Rightarrow 1-{\text{f}}^{\text{'}}\left(\text{x}\right)>0$ nên g(x) đồng biến trên khoảng $(0;2).\underset{x\to 0}{\lim }g\left(x\right)=\frac{1}{2}[0-f(0\left)\right]=\frac{1}{2}f\left(0\right);\underset{x\to 2}{\lim }g\left(x\right)=\frac{1}{2}[2-f(2\left)\right]=1-\frac{1}{2}f\left(2\right)$