If x, y, w, z >0 and $x^4$+$y^4$+$w^4$+$z^4$ <=4 prove 1/$x^4$+1/$y^4$+1/$w^4$+1/$z^4$>=4 I would appreciate suggestions to solve:
If x, y, w, z > 0 and $x^4$ + $y^4$ + $w^4$ + $z^4$ <=4 prove the following:
1/$x^4$ + 1/$y^4$ + 1/$w^4$ + 1/$z^4$ >= 4
From plugging in numbers into Excel, it looks like x, y, w, z must be numbers near 1.
I tried to do a simple case:
If a, b, k > 0 and a + b < k prove the following: 1/a + 1/b > k
a + b < k so 1/(a+b) > 1/k
But 1/a + 1/b > 1/(a+b) so 1/a + 1/b > 1/k which is not what I want to prove.
It seems that I would have to impose a condition for the possible values of k.

Thank you very much for the hints. Here is the solution from the hint of achille hui:
We are told that $x^4$ + $y^4$ + $w^4$ + $z^4$ <=4 so ($x^4$+$y^4$ + $w^4$ + $z^4$)/4 <= 1
But AM >= HM (or HM <= AM) so:
4/(1/$x^4$ + 1/$y^4$ + 1/$w^4$ + 1/$z^4$) <= ($x^4$+$y^4$ + $w^4$ + $z^4$)/4 <= 1
Which means:
4/(1/$x^4$ + 1/$y^4$ + 1/$w^4$ + 1/$z^4$) <= 1
Re-arranging:
4 <= 1/$x^4$ + 1/$y^4$ + 1/$w^4$ + 1/$z^4$
In other words:
1/$x^4$ + 1/$y^4$ + 1/$w^4$ + 1/$z^4$ >= 4
