6. (10 pts) Let $g:(0,\infty)\to\mathbb{R}$ be a twice differentiable function. Assume that

$$g(1)=1,\ g'(1)=3,\ g''(1)=-4.$$

Define a real valued function $h$ on $\mathbb{R}^3 \setminus \{(0,0,0)\}$ by

$$h(x,y,z)=g\left(\sqrt{x^2+y^2+z^2}\right)$$

Calculate $\dfrac{\partial^2 h}{\partial x^2}(P)+\dfrac{\partial^2 h}{\partial z^2}(P)+\dfrac{\partial^2 h}{\partial z^2}(P)$ where $P=\left(\dfrac{2}{3}, \dfrac{2}{3}, -\dfrac{1}{3}\right)$.

提醒：官方原題此處有筆誤，實際上應為 $\dfrac{\partial^2 h}{\partial x^2}(P)+\dfrac{\partial^2 h}{\partial y^2}(P)+\dfrac{\partial^2 h}{\partial z^2}(P)$。