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Question Number 62895 by Tawa1 last updated on 26/Jun/19

Commented by Tony Lin last updated on 26/Jun/19

$$a^2+b^2+c^2=\frac{5}{2}$$$$\Rightarrow a=\pm \sqrt{\frac{1}{2}},b=\pm \sqrt{\frac{1}{2}},c=\pm \sqrt{\frac{3}{2}}$$$$\because \mid ab\mid <\mid bc\mid =\mid ac\mid$$$$\therefore min(ab+bc+ac)$$$$\Rightarrow whenab>0,bc<0,ac<0$$$$min(ab+bc+ac)=\frac{1}{2}-\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2}=\frac{1-2\sqrt{3}}{2}$$

Commented by Tawa1 last updated on 26/Jun/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Answered by Kunal12588 last updated on 26/Jun/19

$$\left(2\right)-\left(1\right)$$$$c^2-a^2=1$$$$c^2+a^2=2$$$$\Rightarrow c^2=\frac{3}{2},a^2=\frac{1}{2}$$$$\Rightarrow c=\pm \frac{\sqrt{6}}{2},a=\pm \frac{\sqrt{2}}{2}$$$$\Rightarrow b=\pm \frac{\sqrt{2}}{2}$$$$ab+bc+ca=k$$$$a<0,b<0,c<0\Rightarrow k=max$$$$a<0,b<0,c>0\Rightarrow k=\frac{1}{2}-\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2}=0.5-1.732=-1.232$$$$a<0,b>0,c>0\Rightarrow k=-\frac{1}{2}+\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2}=-0.5$$$$a<0,b>0,c<0\Rightarrow k=-\frac{1}{2}-\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}=-0.5$$$$a>0,b<0,c<0\Rightarrow k=-\frac{1}{2}+\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2}=-0.5$$$$a>0,b<0,c>0\Rightarrow k=-\frac{1}{2}-\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}=-0.5$$$$a>0,b>0,c<0\Rightarrow k=\frac{1}{2}-\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2}=-1.232$$$$a>0,b>0,c>0\Rightarrow k=max$$$$min(ab+bc+ca)=\frac{1-2\sqrt{3}}{2}$$

Commented by Tawa1 last updated on 26/Jun/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

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