If A, G & H are respectively the AM, GM and HM of three positive numbers a, b & c then the equation whose roots are a, b & c is given by

x^{3} - 3 A x^{2} + 3 G^{3} x - G^{3} = 0

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x^{3} - 3 A x^{2} + 3 (G^{3} / H) x - G^{3} = 0

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x^{3} + 3 A x^{2} + 3 (G^{3} / H) x - G^{3} = 0

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x^{3} - 3 A x^{2} + 3 (G^{3} / H) x + G^{3} = 0

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Solution

The correct option is B $x^{3} - 3 A x^{2} + 3 (G^{3} / H) x - G^{3} = 0$
$A = a + b + c / 3$
$G = \sqrt[3]{a b c}$
$H = \frac{3}{1 / a + 1 / b + 1 / c}$
$a + b + c = 3 A, a b + b c + c a = {3 G}^{3} / H, a b c = G^{3}$
so the equation is $x^{3} - 3 A x^{2} + 3 (\frac{G^{3}}{H}) x - G^{3} = 0$

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