a) LHS follows from: $x^4+y^4 \ge \frac{2}{9}(x^2-xy+y^2) \Leftrightarrow 4(x+y)^2(x^2-xy+y^2+3(x-y)^2(x+y)^2 \ge 0$ Equality holds when $x=-y= \pm \frac{1}{\sqrt{3}}$ RHS follows from $x^4+y^4 \le 2(x^2-xy+y^2)^2 \Leftrightarrow (x-y)^4 \ge 0$ Equality holds when $x=y=\sqrt{2}$ b) It follows from power mean inequality and LHS of the a) part: $ \frac{x^{2n}+y^{2n}}{2} \ge \left( \frac{x^4+y^4}{2} \right)^{\frac{2n}{4}} \ge \frac{1}{3^n}$