$(10$ points$)$ Let $f(x,y)=x^4+xy+y^4+1.$

$($a$)(5$ points$)$ Find and classify the critical points of $f$ as local maxima, local minima, or saddle points.

$($b$)(5$ points$)$ Find the absolute maximum and the minimum values of $f(x,y)$ on the region defined by $x^4+y^{4}1.$