A plane passing through $$\left( {1,\,1,\,1} \right)$$   cuts positive direction of coordinate axes at $$A,\,B$$  and $$C$$, then the volume of tetrahedron $$OABC$$   satisfies :

Solution :
$$\eqalign{ & {\text{Let the equation of the plane be}} \cr & \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \cr & \Rightarrow \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1 \cr & \Rightarrow {\text{Volume of tetrahedron}}\,OABC = V = \frac{1}{6}\left( {abc} \right) \cr & {\text{Now, }}{\left( {abc} \right)^{\frac{1}{3}}} \geqslant \frac{3}{{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}} \geqslant 3\left( {{\text{G}}{\text{.M}}{\text{.}} \geqslant {\text{H}}{\text{.M}}{\text{.}}} \right) \cr & {\text{or, }}abc \geqslant 27 \Rightarrow V \geqslant \frac{9}{2} \cr} $$