Linear Transformations from Eigenvalues and Eigenvectors 4 Consider the linear transformation $f: V → V$ given by 
$$f(u) = \frac{-d^2u (x)}{dx^2}$$ 
Where $V$ is the vector space of all continuously differentiable functions on the interval $0\le x \le L$ with $L$ a positive number. Find the possible eigenvalues $\
\lambda$ of $f(u) = \lambda u$ subject to the condition that $ u(x=0)=u(x =L)=0$. For the smallest two eigenvalues find the corresponding eigenfunction $u(x)$ normalized such that
$$\int_0^L dx |u(x)|^2 = 1$$ 
I have no idea what to do. I know I'm supposed to split it up into three parts but I can't seem to make sense of it. Can someone please give me some guidance? I've attached a picture of the actual problem here.

The eigenfunctions are those functions such that $Vu = \lambda u$
and $V=-\frac{d^2u}{dx^2}$
$u'' = -\lambda u$
There are only a few forms u can take on.
$u= e^{kx}, u= e^{-kx},u= \sin(kx), u= \cos(kx)$
where $k = \sqrt{|\lambda|}$
Next condition $u(0) = u(L) = 0,  u$ is a sin function.
$u = \sin(\frac{n\pi}{L} x)$
I am going to leave the rest to you.
