Finding a bounded functional with no global extrema

Question

I need to find a bounded functional with no extrema in the space $X:=\{ y\in C^1[-1,1]:y(-1)=y(1)=0\}$
My first guess was $I(y)=\int_{-1}^1\cos(y(x))dx$ which is indeed bounded but has a global maximum at $y(x)=0$. I think that if I tried this aproach with another bounded integrand I'd run into the same problem. What else can I try?