If $x y = 6$ , then minimum value of $2 x + 3 y$ is __________.

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Solution

By applying the AMGM property,
$\frac{2 x + 3 y}{2} \geq \sqrt{(2 x) (3 y)}$
$\Rightarrow 2 x + 3 y \geq 2 \sqrt{6 x y}$
$\Rightarrow 2 x + 3 y \geq 2 \sqrt{6 (6)}$
$\Rightarrow 2 x + 3 y \geq 2 \times 6$
$\Rightarrow 2 x + 3 y \geq 12$
Hence, minimum value of $2 x + 3 y$ will be $12$ .

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