Find the smallest integer n?

Find the smallest integer n such that

$$(x^2 + y^2 + z^2)^2 \leq n(x^4 + y^4 + z^4)$$ for all real numbers x, y, and z.

1 Answer
George C.
May 14, 2017

#n=3#

Explanation:

Note that:

#0 <= (x^2-y^2)^2+(y^2-z^2)^2+(z^2-x^2)^2#

#color(white)(0) = 2x^4+2y^4+2z^4-2x^2y^2-2y^2z^2-2z^2x^2#

#color(white)(0) = 3(x^4+y^4+z^4)-(x^4+y^4+z^4+2x^2y^2+2y^2z^2+2z^2x^2)#

#color(white)(0) = 3(x^4+y^4+z^4)-(x^2+y^2+z^2)^2#

So #n=3# works.

To see that no smaller value of #n# works...

Let #x = y = z = 1#

Then we have:

#(x^2+y^2+z^2)^2 = 3^2 = 9 = 3(x^4+y^4+z^4)#

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