First slide

Means - A.M/G.M/H.M

Question

If a, b, c are positive then the minimum value of $a^{\log b - \log c} + b^{\log c - \log a} + c^{\log a - \log b}$ is

Moderate

A

3
B

1
C

9
D

16

Solution

Since A.M. ≥ G.M.
$\begin{array}{l} ∴ \frac{a^{\log b - \log c} + b^{\log c - \log a} + c^{\log a - \log b}}{3} \geq \sqrt[3]{a^{\log b - \log c} \cdot b^{\log c - \log a} \cdot c^{\log a - \log b}} \\ (1) \\ Let x = a^{\log b - \log c} \cdot b^{\log c - \log a} \cdot c^{\log a - \log b} \\ \Rightarrow \log x = (\log b - \log c) \log a + (\log c - \log a) \log b + (\log a - \log b) \log c \\ \Rightarrow \log x = 0 \Rightarrow x = 1 \\ ∴ From (1), \\ a^{\log b - \log c} + b^{\log c - \log a} + c^{\log a - \log b} \geq 3 . \end{array}$

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