If $a b = 2 a + 3 b, a > 0, b > 0$ then the minimum value of $a b$ is

12

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24

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\frac{1}{4}

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none of these

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Solution

The correct option is C $24$
Using A.M $\geq$ G.M
$\frac{(2 a + 3 b)}{2}$ $\geq$ $\sqrt{6 a b}$ given $2 a + 3 b = a b$
So $\frac{a b}{2} \geq \sqrt{6 a b} \Rightarrow a b \geq 24$
So the minimum value is $24$

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If ab=2a+3b,a>0,b>0 then the minimum value of ab is

The correct option is C 24Using A.M ≥ G.M(2a+3b)2 ≥ √6ab given 2a+3b=abSo ab2≥√6ab⇒ab≥24So the minimum value is 24

If $a b = 2 a + 3 b, a > 0, b > 0$ then the minimum value of $a b$ is

The correct option is C $24$
Using A.M $\geq$ G.M
$\frac{(2 a + 3 b)}{2}$ $\geq$ $\sqrt{6 a b}$ given $2 a + 3 b = a b$
So $\frac{a b}{2} \geq \sqrt{6 a b} \Rightarrow a b \geq 24$
So the minimum value is $24$