If $x y = a^{2}$ and $S = b^{2} x + c^{2} y$ where a,b and c are constants then the minimum value of S is

a b c

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b c \sqrt{a}

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2 a b c

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none of these

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Solution

The correct option is C $2 a b c$
Given $x y = a^{2}$ and $S = b^{2} x + c^{2} y$
$\Rightarrow S = b^{2} x + c^{2} a^{2} / x$
$\Rightarrow \frac{d S}{d x} = b^{2} - c^{2} a^{2} / x^{2}$
For maximum or minimum value of $S$
$\frac{d S}{d x} = 0 = b^{2} - c^{2} a^{2} / x^{2} \Rightarrow x = \pm a c / b$
Now $\frac{d S}{d x} = 2 c^{2} a^{2} / x^{3}$
Clearly at $x = a c / b$ , $\frac{d S}{d x} = 2 b^{3} / a c > 0$ (Assuming that $b^{3} / a c > 0$ )
Hence minimum value of $S$ is $= b^{2} (a c / b) + c^{2} (b / a c) = 2 a b c$

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