Tough proof question! Prove that 4xyzw <= x^4 + y^4 + z^4 + w^4?

Question

It is related to the AGM Inequality and is supposed to be used when doing this. The AGM inequality is

2xy <= x^2 + y^2

Answer 1

Geometric mean of the powers: xyzw
Arithmetic mean of the powers: (1/4)(x^4+y^4+z^4+w^4)

Geometric mean <= Arithmetic mean implies

xyzw<= (1/4)(x^4+y^4+z^4+w^4) as required.

Answer 2

I think it will be apparent if you first deal with the AGM inequality. If you subtract 2xy from both sides, you can express it as
0<= (x2-xy)+(y2-xy) or x(x-y)+y(y-x)
If y=x, the result is zero. If either is larger than the other, the result is greater than zero.

You can proceed the same way with the given problem.