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Question Number 157562 by Ar Brandon last updated on 24/Oct/21

Answered by FongXD last updated on 24/Oct/21

$$\mathrm{Given}:\mathrm{x}+\mathrm{y}+\mathrm{z}=3$$$$\mathrm{square}\mathrm{both}\mathrm{sides}\mathrm{of}\mathrm{the}\mathrm{equation}$$$$\iff {\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2+2\mathrm{xy}+2\mathrm{yz}+2\mathrm{xz}=9$$$$\mathrm{AM}-\mathrm{GM}\mathrm{inequality}:\mathrm{a}+\mathrm{b}⩾2\sqrt{\mathrm{ab}}$$$$\mathrm{we}\mathrm{get}:{\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2+({\mathrm{x}}^2+{\mathrm{y}}^2)+({\mathrm{y}}^2+{\mathrm{z}}^2)+({\mathrm{x}}^2+{\mathrm{z}}^2)⩾9$$$$\Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2⩾3$$

Commented by Ar Brandon last updated on 24/Oct/21

$$\mathrm{thanks}$$

Answered by mnjuly1970 last updated on 24/Oct/21

$$c-sinequality$$$${(x+y+z)}^2⩽(1+1+1)(x^2+y^2+z^2)$$$$3⩽x^2+y^2+z^2$$

Commented by Ar Brandon last updated on 24/Oct/21

$$\mathrm{thank}\mathrm{you}\mathrm{sir}.$$

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