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Omni-MATH / Let m,n,p be fixed positive real numbers which satisfy mnp=8. Depending on these constants, find the minimum of x²+y²+z²+mxy+nxz+pyz, where x,y,z are arbitrary positive real number…

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problem

Let m,n,pm, n, p be fixed positive real numbers which satisfy mnp=8mnp = 8. Depending on these constants, find the minimum of x2+y2+z2+mxy+nxz+pyz,x^2+y^2+z^2+ mxy + nxz + pyz, where x,y,zx, y, z are arbitrary positive real numbers satisfying xyz=8xyz = 8. When is the equality attained? Solve the problem for: [list=a][*]m=n=p=2,m = n = p = 2, [*] arbitrary (but fixed) positive real numbers m,n,p.m, n, p.[/list]
Plain-text mathematical notation (without MathML)
Let m,n,p be fixed positive real numbers which satisfy mnp=8. Depending on these constants, find the minimum of x²+y²+z²+mxy+nxz+pyz,
where x,y,z are arbitrary positive real numbers satisfying xyz=8. When is the equality attained?
Solve the problem for:
[list=a][*]m=n=p=2,
[*] arbitrary (but fixed) positive real numbers m,n,p.[/list]
Original LaTeX notation
Let $m, n, p$ be fixed positive real numbers which satisfy $mnp = 8$. Depending on these constants, find the minimum of $$x^2+y^2+z^2+ mxy + nxz + pyz,$$
where $x, y, z$ are arbitrary positive real numbers satisfying $xyz = 8$. When is the equality attained?
Solve the problem for:
[list=a][*]$m = n = p = 2,$
[*] arbitrary (but fixed) positive real numbers $m, n, p.$[/list]

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Source and history

task_id
5f55a6a5-aab1-5a4f-9126-4774535c9040
task_key
test--5f55a6a5-aab1-5a4f-9126-4774535c9040
task_revision_id
1
config / split
/ test

Official source

initial import