Question

If three positive real numbers $a,b,c$ are in $AP$ and $abc=4$, then the minimum possible value of $b$ is

• A. $2^{3/2}$
• B. $2^{2/3}$
• C. $2^{1/3}$
• D. $2^{5/2}$

Answer 1

Answer: (B)

Since, $a$, $b$ and $c$ are in $AP$

Let $d$ be the common difference

$\therefore a=b-d,b=d,c=b+d$

Also, $abc=4$

$\Rightarrow \left(b-d\right)d\left(b+d\right)=4$

$\Rightarrow \left(b^2-d^2\right)b=4$

$\Rightarrow b^3=4+d^{2}b$

$\Rightarrow b^3\ge 4$

$\Rightarrow b\ge {\left(2\right)}^{2/3}$