Question 20
$K$ means the maximum value of $$
\left|x_{1}-x_{2}\right| \cdot\left|x_{1}-x_{3}\right| \cdot\left|x_{1}-x_{4}\right| \cdot\left|x_{2}-x_{3}\right| \cdot\left|x_{2}-x_{4}\right| \cdot\left|x_{3}-x_{4}\right|
$$ or $x_{1}, x_{2}, x_{3}, x_{4} \in [0 ; 1]$.
a. Prove that $\frac{4}{243} \lt K \lt \frac{1}{27}$.
b. Determine $K$.
\left|x_{1}-x_{2}\right| \cdot\left|x_{1}-x_{3}\right| \cdot\left|x_{1}-x_{4}\right| \cdot\left|x_{2}-x_{3}\right| \cdot\left|x_{2}-x_{4}\right| \cdot\left|x_{3}-x_{4}\right|
$$ or $x_{1}, x_{2}, x_{3}, x_{4} \in [0 ; 1]$.
a. Prove that $\frac{4}{243} \lt K \lt \frac{1}{27}$.
b. Determine $K$.