Answer
The Laplace's equation is satisfied.
Work Step by Step
We need to compute the Laplace's equation and prove that it equals to $0$.
In order to find the partial derivative, we will differentiate with respect to $x$, by keeping $y$ and $z$ as a constant, and vice versa:
$f_x=-6xz$; $f_y=-6yz $ and $f_z=6z^2-3x^2-3y^2$
Consider the Laplace's equation $\nabla^2 f =\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}+\dfrac{\partial^2 f}{\partial z^2}$
$\nabla^2 f =-6z-6z+12z=0$
Thus, the Laplace's equation is satisfied.
