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Question Number 32768 by Rasheed.Sindhi last updated on 01/Apr/18

$$\mathrm{Prove}\mathrm{that}{a}^2+b^2+c^2⩾ab+bc+ca$$$$\mathrm{\forall }a,b,c\in \mathbb{R}$$

Answered by mrW2 last updated on 02/Apr/18

$$a^2+b^2-2ab={(a-b)}^2⩾0$$$$a^2+b^2⩾2ab$$$$b^2+c^2⩾2bc$$$$c^2+a^2⩾2ca$$$$\mathrm{\Sigma }:$$$$2(a^2+b^2+c^2)⩾2(ab+bc+ca)$$$$a^2+b^2+c^2⩾ab+bc+ca$$

Commented by Joel578 last updated on 02/Apr/18

$$\mathrm{In}3\mathrm{rd}\mathrm{line},\mathrm{did}\mathrm{you}\mathrm{mean}$$$$c^2+a^2⩾2ca?$$

Commented by Rasheed.Sindhi last updated on 02/Apr/18

$$\mathrm{Th}\mathrm{\forall }\mathrm{n}\mathbb{k}\mathrm{s}\mathrm{Sir}!$$

Commented by mrW2 last updated on 02/Apr/18

$$yes.thanks!$$

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