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?