  We prove that if every bounded linear operator (or $N$-homogeneous
polynomials) with the compact approximation property attains its numerical
radius, then $X$ is a finite dimensional space. Moreover, we present an
improvement of the polynomial James' theorem for numerical radius proved by
Acosta, Becerra Guerrero and Gal$\'a$n in 2003. Finally, the denseness of
weakly (uniformly) continuous $2$-homogeneous polynomials on a Banach space
whose Aron-Berner extensions attain their numerical radii is obtained.
