Question

If $ab=2a+3b$ where a > 0, b > 0, then find the minimum value of ab.

Answer 1

$ab=2a+3b$ ...(1)

Using $AM\ge GM$

$\frac{(2a+3b)}{2}\ge {\left(6ab\right)}^{\frac{1}{2}}\Rightarrow 2a+3b\ge 2{\left(6ab\right)}^{\frac{1}{2}}$ from(1)

$\Rightarrow ab\ge 2{\left(6ab\right)}^{\frac{1}{2}}\Rightarrow {\left(ab\right)}^{\frac{1}{2}}\ge 2{\left(6\right)}^{\frac{1}{2}}$

$\Rightarrow ab\ge 24$