The question:
Prove that $4xyzw \leq x^4+y^4+z^4+w^4$ for any $x, y, z, w \in ℝ$.
I am sure there are various tricks that can be used to prove this inequality, however, I can't seem to think of them. Perhaps I'm overthinking it.
What I have tried doing:
Tackling the inequality like so (breaking it down into two pieces and adding them back together):
$(x^4+y^4)\ge 2x^2y^2$
$(z^4+w^4)\ge 2z^2w^2$
$(x^4+y^4) + (z^4+w^4)\ge 2x^2y^2 + 2z^2w^2$.
Well, this isn't really what I need.
I am quite lost, and would greatly appreciate any kind of help!
Now, $$2x^2y^2+2z^2w^2=2(xy-zw)^2+4xyzw\geq4xyzw.$$