Question

If a, b and c are positive real number then a/b + b/c + c/a is greater than or equal to

Answer 1

Dear student

$$\begin{gathered} \mathrm{Applying}\mathrm{relation}\mathrm{A}.\mathrm{M}\ge \mathrm{G}.\mathrm{M} \\ \frac{{\displaystyle \frac{\mathrm{a}}{\mathrm{b}}}+{\displaystyle \frac{\mathrm{b}}{\mathrm{c}}}+{\displaystyle \frac{\mathrm{c}}{\mathrm{a}}}}{3}\ge \sqrt[3]{\frac{\mathrm{a}}{\mathrm{b}}\times \frac{\mathrm{b}}{\mathrm{c}}\times \frac{\mathrm{c}}{\mathrm{a}}}\left[\because \mathrm{A}.\mathrm{M}=\frac{{\mathrm{a}}_1+{\mathrm{a}}_2+...+{\mathrm{a}}_{\mathrm{n}}}{\mathrm{n}}\mathrm{and}\mathrm{G}.\mathrm{M}=\sqrt[\mathrm{n}]{{\mathrm{a}}_1\times {\mathrm{a}}_2\times ...\times {\mathrm{a}}_{\mathrm{n}}}\right] \\ \Rightarrow \frac{{\displaystyle \frac{\mathrm{a}}{\mathrm{b}}+\frac{\mathrm{b}}{\mathrm{c}}+\frac{\mathrm{c}}{\mathrm{a}}}}{3}\ge 1 \\ \Rightarrow \frac{\mathrm{a}}{\mathrm{b}}+\frac{\mathrm{b}}{\mathrm{c}}+\frac{\mathrm{c}}{\mathrm{a}}\ge 3 \end{gathered}$$
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