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=3e^{3x+4y} \cos (5z) \\ f_y=4e^{3x+4y} \cos (5z)
\\ f_z=-5e^{3x+4y} \sin (5z) $
Consider the Laplace's equation $\nabla^2 f =\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}+\dfrac{\partial^2 f}{\partial z^2}$
$\nabla^2 f =3 \cos 5 z \cdot e^{3x+4y} \cdot 3+4 \cos 5 z \cdot e^{3x+4y} \cdot 4+(-5) \cos 5 z \cdot e^{3x+4y} \cdot 5=9\cos 5 z \cdot e^{3x+4y}+16\cos 5 z \cdot e^{3x+4y}-25\cos 5 z \cdot e^{3x+4y}=0$
Thus, the Laplace's equation is satisfied.
