Heat Equation with Neumann Boundary conditions

So essentially, I am given these conditions:

Satisfy laplace's equation inside a rectangle for

the problem comes in three parts:

1) without solving this problem, briefly explain the physical condition underwhich there is a solution to this problem.

*I think I understand this one. so you can have a steady state, which is necessary when solving a heat equation with the Laplace equation.*

2) Solve this problem by the method of separation of variables. Show that the method works only under the condition of part (1).

*I can solve the problem easily (as far as PDE's go, it's not too bad). But I don't have the theory knowledge to answer the second part. I don't know how to mathematically prove that it is necessary for the integral of F(x) to be zero over (0,L)*

The solution I get is:

[tex]U(x,y)= \sum_{n=1}^{\infty} A_n\cos(\frac{nx\pi}{L})\sinh(\frac{ny\pi}{L})[/tex]

[tex]A_n= \frac{2}{L\sinh(nH\pi/L)}\int_{0}^{L}f(x)\cos(\frac{nx\pi}{L})dx[/tex]

3)The solution (part 2) has an arbitrary constant. Determine it by consideration of the time dependent heat equation subject to the initial condition:

U(x,y,0)=g(x,y) (the 0 is t=0)

*I have no clue how to start this at all. help!*