Question

If three positive real numbers $a,b,c$ are in A.P. and $abc=4$, then minimum possible value of $b$ is _______

Answer 1

If three numbers a,b,c are in A.P. then $2b=a+c$
As we know that $A.M.\ge G.M.$ of a and c.
It is given that $abc=4$
$\Rightarrow \frac{a+c}{2}\ge \sqrt{ac}$
$\Rightarrow b\ge \sqrt{ac}$
Multiply on both sides with $\sqrt{b}$ then we get,
$\Rightarrow b^{\frac{3}{2}}\ge \sqrt{abc}$
$\Rightarrow b^{\frac{3}{2}}\ge \sqrt{4}=2\left[\text{putting abc = 4}\right]$
$\Rightarrow b\ge 2^{\frac{2}{3}}=4^{\frac{1}{3}}$
Hence, minimum value of $b$ is $4^{\frac{1}{3}}.$