WWJD (What would Jacobi do?)

The total derivative or Jacobian, $J$ , of a function is defined as the matrix with the partial derivatives of the component functions of $f$ .

$\large f(x) = \begin{bmatrix} {f}_{1} \\ {f}_{2} \\ {f}_{3} \end{bmatrix} \Longrightarrow J = \begin{bmatrix} \frac{\partial {f}_{1}}{\partial x} & \frac{\partial {f}_{1}}{\partial y} & \frac{\partial {f}_{1}}{\partial z} \\ \frac{\partial {f}_{2}}{\partial x} & \frac{\partial {f}_{2}}{\partial y} & \frac{\partial {f}_{2}}{\partial z} \\ \frac{\partial {f}_{3}}{\partial x} & \frac{\partial {f}_{3}}{\partial y} & \frac{\partial {f}_{3}}{\partial z} \end{bmatrix}$

$\large \Longrightarrow J = \begin{bmatrix} 1 & 1 & 1 \\ y + z & x + z & x + y \\ yz & xz & xy \end{bmatrix}$

$\large \Longrightarrow det(J) = xy(x - y) + yz(y - z) + zx(z - x)$

$\large = -(x - y)(y - z)(z - x)$ .

Thus, the determinant is zero whenever any two of the components of the point are equal which is true for every point given.

Therefore the answer is $\boxed{0}$ .