orl wrote: Let \[ f(x_1,x_2,x_3) = -2 \cdot (x_1^3+x_2^3+x_3^3) + 3 \cdot (x_1^2(x_2+x_3) + x_2^2 \cdot (x_1+x_3) + x_3^2 \cdot ( x_1+x_2 ) - 12x_1x_2x_3. \] For any reals $r,s,t$, we denote \[ g(r,s,t)=\max_{t\leq x_3\leq t+2} |f(r,r+2,x_3)+s|. \] Find the minimum value of $g(r,s,t)$. $ f(x_1,x_2,x_3)=(x_1+x_2-2x_3)(x_1-2x_2+x_3)(-2x_1+x_2+x_3) $ $ |f(r,r+2,x)+s|=|2y^3-18y+s| $, $ y=1+r-x $ $ g(r,s,t)=\max_{r-t-1\leq y\leq r-t+1} |2y^3-18y+s| $ Let $h(x)=2x^3-18x$. $ \min_{r,s,t\in\mathbb{R}} g(r,s,t)=\frac{1}{2}\min_{a\in\mathbb{R}} \left(\max_{a\leq x\leq a+2} h(x)-\min_{a\leq x\leq a+2} h(x)\right)=6\sqrt{3}-\frac{20\sqrt{6}}{9} $