Answer
The Laplace's equation is satisfied.
Work Step by Step
We need to verify the Laplace's equation $\nabla^2 f =\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}=0$
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=3e^{3x+4y} \cos (5z) ; \\ f_y=4e^{3x+4y} \cos (5z) ;
\\ f_z=-5e^{3x+4y} \sin (5z) $
Now, $\nabla^2 f =\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}+\dfrac{\partial^2 f}{\partial z^2} =3 \cos 5 z \cdot e^{3x+4y} \cdot 3+4 \cos 5 z \cdot e^{3x+4y} \cdot 4+(-5) \cos 5 z \cdot e^{3x+4y} \cdot 5$
or, $\nabla^2 f =9\cos 5 z \cdot e^{3x+4y}+16\cos 5 z \cdot e^{3x+4y}-25\cos 5 z \cdot e^{3x+4y}=0$
So, the Laplace's equation is satisfied.