A) \[3\sqrt{3}\]
B) \[\sqrt{3}\]
C) \[2\sqrt{2}\]
D) None of these
Correct Answer: B
Solution :
[b] \[\sqrt{uv}\le \frac{u+v}{2}\] \[(G.M.\le A.M.)\] \[\Rightarrow \,\,\,\,\,\,\,\,w\sqrt{uv}\le w\left( \frac{u+v}{2} \right)\] Similarly, \[u\sqrt{vw}\le u\left( \frac{v+w}{2} \right)\] And \[v\sqrt{wu}\le v\left( \frac{w+u}{2} \right)\] Adding above three inequalities, we get \[uv+vw+wu\ge u\sqrt{vw}+v\sqrt{wu}+w\sqrt{uv}\ge 1\] Also, we have \[{{(u+v+w)}^{2}}={{u}^{2}}+{{v}^{2}}+{{w}^{2}}+2(uv+vw+wu)\] \[\ge 3(uv+vw+wu)\ge 3\] \[(\,\,\because \,\,\,\,\,{{u}^{2}}+{{v}^{2}}+{{w}^{2}}\ge uv+vw+wu)\]
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