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If-x-y-z-R-and-x-2-y-2-z-2-3-Prove-that-1-4-x-1-4-y-1-4-z-1-




Question Number 210571 by hardmath last updated on 12/Aug/24
If  x,y,z∈R^+   and  x^2 +y^2 +z^2 =3  Prove that  (1/(4−x))  +  (1/(4−y))  +  (1/(4−z))  ≤  1
Ifx,y,zR+andx2+y2+z2=3Provethat14x+14y+14z1
Answered by A5T last updated on 15/Aug/24
≡(1/(x−4))+(1/(y−4))+(1/(z−4))≥−1  (1/(x−4))+(1/(y−4))+(1/(z−4))≥(9/(x+y+z−12))  1=(√((x^2 +y^2 +z^2 )/3))≥((x+y+z)/3)⇒x+y+z≤3  ⇒(9/(x+y+z−12))≥(9/(−9))=−1  ⇒(1/(4−x))+(1/(4−y))+(1/(4−z))≤1
1x4+1y4+1z411x4+1y4+1z49x+y+z121=x2+y2+z23x+y+z3x+y+z39x+y+z1299=114x+14y+14z1

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