If $\mathrm{x},\mathrm{y},\mathrm{z}\in [0,1]$ then

${\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2\le \mathrm{xyz}+\mathrm{k}$.

Then find the least positive integral value of k is______

Recall that a quadratic function with positive leading coefficient on any interval $[\mathrm{\alpha },\mathrm{\beta }]$ attains a maximum at either one of the endpoints. Now, consider the function $\mathrm{f}(\mathrm{x},\mathrm{y},\mathrm{z})={\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2-\mathrm{xyz}-2$. This is quadratic in each of $\mathrm{x},\mathrm{y},\mathrm{z}$, which implies that the maximum of $f$is obtained when $\mathrm{x},\mathrm{y},\mathrm{z}\in 0,1$. It is easy to check cases to see that the inequality holds.