integration by parts problem

Suppose that f(x) and g(x) are functions with f(0)=g(0)=0, f(1)=g(1)=0, and with continuous second derivatives.

Use integration by parts to show that

Then come up with specific examples for f(x) and g(x) that satisfy the above.

I tried using

u=g(x) v=f(x)

du=g'(x)dx dv=f''(x)dx

then

u=g'(x) v=f(x)

du=g''(x)dx dv=f'(x)dx

but I ended up with

and I don't know how to get rid of the two middle functions...

if I switched what U was for the second time I applied it, I get back to where I began and If I apply IBP again, i get into 3rd derivatives...

I'm really confused. Can someone give me some advice?