Question

find the maximum $p$ such that $2x^4y^2 9y^4z^2 12z^4x^2 - px^2y^2z^2$ is always nonnegative for all real $x$, $y$, and $z.$

Answer 1

So now you just need to use LaGrange multipliers to minimize the function of a unit sphere (a compact set) and work out for which values of p the minimum is non-negative.

2a²b+9b²c+12c²a≥pabc

for a,b,c≥0.Letting a=3k,b=2m,c=n yields

36(k²m+m²n+n²k)≥(6p)kmn

k²m+m²n+n²k≥(p6/5)kmn

We now use AM-GM on k2m,m2n,n2k to get that

k²m+m²n+n²k/3 ≥ (k³m³n³)∧13

k2m+m2n+n2k≥3kmn

So we know that this applies to all p18. Furthermore, taking k=m=n turns the inequality into equality, therefore any p>18 fails. As a result, p=18 is the maximum.

Call that function f, and keep in mind that f(x,y,z)=(x,y,z) for all >0 and x,y,zR. So all you have to do is ensure that f is non-negative on the unit sphere.

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