<s>[INST] How can I prove this inequality:$(x+y)^4\le8(x^4+y^4)$? I have to proce that $(x+y)^4\le8(x^4+y^4)$
I have seen that the equality is when $x=y$... 
I have tried to develop $(x+y)^4$ but it leads me nowhere... I think that I have to pass by the intermediate of the averages, but I don't see witchone... Could someone please help?
 [/INST] $$(x+y)^4= x^4+4x^3y+6x^2y^2+4xy^3+y^4$$
Now, by AM-GM you have
$$
x^3y \leq \frac{x^4+x^4+x^4+y^4}{4} \\
x^2y^2 \leq \frac{x^4+y^4}{2} \\
xy^3\leq \frac{x^4+y^4+y^4+y^4}{4}$$
Thus
$$(x+y)^4 \leq x^4+4 \cdot \frac{x^4+x^4+x^4+y^4}{4} +6 \frac{x^4+y^4}{2} +4 \frac{x^4+y^4+y^4+y^4}{4} +y^4$$
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