Question

If $a^{x_1}=b^{x_2}=c^{x_3}$ where $x_1,x_2,x_3$ are unequal positive numbers and a, b, c are in G.P., then $x_1^3+x_3^3$ is

• A. $\ge 2x_2^3$
• B. $\le 2x_2^3$
• C. $>2x_2^3$
• D. None of these

Answer 1

The correct option is C $>2x_2^3$

Let $a^{x_1}=b^{x_2}=c^{x_3}=k$ (say) where $x_1,x_2,x_3$ are unequal positive numbers.
$\Rightarrow a=k^{\frac{1}{x_1}},b=k^{\frac{1}{x_2}},c=k^{\frac{1}{x_3}}$
Also, $a,b,c$ are in G.P.
$\Rightarrow b^2=ac$
$\Rightarrow k^{\frac{2}{x_2}}=k^{\frac{1}{x_1}}\cdot k^{\frac{1}{x_3}}$
$\Rightarrow k^{\frac{2}{x_2}}=k^{\frac{1}{x_1}+\frac{1}{x_3}}$
$\Rightarrow \frac{2}{x_2}=\frac{1}{x_1}+\frac{1}{x_3}$
$\Rightarrow x_1,x_2,x_3$ are in H.P.
Therefore, $x_2$ is the H.M. of $x_1$ and $x_3$.
Since, $x_1$ and $x_3$ are unequal.
Therefore, G.M of $x_1,x_3$$>$H.M of $x_1,x_3$
$\Rightarrow \sqrt{x_1{x}_3}>x_2$ $...\left(1\right)$
Consider, the positive numbers ${x_1}^3$ and ${x_{3.}}^3$
Also, A.M$>$G.M
$\Rightarrow \frac{{x_1}^3+{x_3}^3}{2}>\sqrt{{x_1}^3{x_3}^3}$
$\Rightarrow {x_1}^3+{x_3}^3>2\sqrt{{x_1}^3{x_3}^3}$
$\Rightarrow {x_1}^3+{x_3}^3>2{\left(\sqrt{x_1{x}_3}\right)}^3$
From $\left(1\right)$, we have $\sqrt{x_1{x}_3}>x_2$
$\Rightarrow {x_1}^3+{x_3}^3>2{x_2}^3$