Answer
The Laplace's equation is satisfied.
Work Step by Step
We need to verify Laplace's equation
$\nabla^2 f =\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}+\dfrac{\partial^2 f}{\partial z^2}$
We take the first partial derivative of the given function $f(x,y,z)$ with respect to $x$, by treating $y$ and $z$ as a constant, and vice versa:
$f_x=2x \\ f_y=2y \\f_z=-4z$
Now, $\nabla^2 f =2+2+(-4)=0$
Thus, the Laplace's equation is satisfied.
