<s>[INST] Minimize $x^4+ \frac{1}{x^4} +y^4+ \frac{1}{y^4}$ 
Minimize $${x^4+ \frac{1}{x^4} +y^4+ \frac{1}{y^4}}  $$
  Subject to $$x^2+y^2=4$$


I've managed to solve this by substituting $x=(2+t)^\frac{1}{2}$ and $y=(2-t)^\frac{1}{2}$ and I arrived at a minimum of $\frac{17}{2}$ when $t=0$, but it's kind of tedious.
So I'm asking if there are straightforward ways of finding the minimum using inequalities and not using symmetry.  
 [/INST] Let $f(x)=x^2+\frac{1}{x^2}$. 
Thus, $f''(x)>0$. 
Thus, by Jensen
$$x^4+\frac{1}{x^4}+y^4+\frac{1}{y^4}\geq2\left(\left(\frac{x^2+y^2}{2}\right)^2+\frac{1}{\left(\frac{x^2+y^2}{2}\right)^2}\right)=\frac{17}{2}.$$
The equality occurs for $x=y=\sqrt2$, which says that we got a minimal value.
Done!
</s>