How big does $r$ need to be, to ensure that $x^4+y^4 >r^2$ for all $x^2+y^2 =r^2$? Consider the circle $x^2+y^2 = r^2$ for some fixed $r>0$. How big does $r$ need to be, to ensure that $$x^4+y^4 >r^4$$ for all $x^2+y^2 =r^2$? I know that $x^2<x^4$ whenever $|x| >1$, but I'm not sure how to use that here.

By C-S $$(1^2+1^2)(x^4+y^4)\geq(x^2+y^2)^2.$$
Thus, $$x^4+y^4\geq\frac{1}{2}r^4.$$
