$$\max _{0 \leq x \leq 2}\left\{|f(x)|,\left|f^{\prime \prime}(x)\right|\right\} \leq 1 ，$$

$$\left\{\begin{array}{l} f(0)=f(x)-x f^{\prime}(x)+\frac{x^2}{2} f^{\prime \prime}\left(\xi_1\right) \\ f(2)=f(x)+(2-x) f^{\prime}(x)+\frac{(2-x)^2}{2} f^{\prime \prime}\left(\xi_2\right), \end{array}\right.$$

$$f(2)-f(0)=2 f^{\prime}(x)+\frac{(2-x)^2}{2} f^{\prime \prime}\left(\xi_2\right)-\frac{x^2}{2} f^{\prime \prime}\left(\xi_1\right) .$$

$$2 f^{\prime}(x)=f(2)-f(0)+\frac{x^2}{2} f^{\prime \prime}\left(\xi_1\right)-\frac{(2-x)^2}{2} f^{\prime \prime}\left(\xi_2\right),$$

\begin{aligned} & 2\left|f^{\prime}(x)\right|=\left|f(2)-f(0)+\frac{x^2}{2} f^{\prime \prime}\left(\xi_1\right)-\frac{(2-x)^2}{2} f^{\prime \prime}\left(\xi_2\right)\right| \\ & \leq|f(2)|+|f(0)|+\frac{x^2}{2}\left|f^{\prime \prime}\left(\xi_1\right)\right|+\frac{(2-x)^2}{2}\left|f^{\prime \prime}\left(\xi_2\right)\right| \\ & \leq 2+\frac{x^2}{2}+\frac{(2-x)^2}{2}=(x-1)^2+3 \leq 1+3=4 \end{aligned}

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