If $a + b + c + d + e + f = 12$ then the maximum value of $a b + b c + c d + d e + e f + f a$ is (a, b, c, d, e, f are non negative real numbers)

36

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24

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30

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none of these

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Solution

The correct option is B $24$
AM $\geq$ GM
$\frac{a + b + c + d + e + f}{6} \geq {(a b c d e f)}^{1 / 6} 2^{2} \geq {(a b c d e f)}^{1 / 3} (1) \frac{a b + b c + c d + d e + e f + f a}{6} \geq {(a b c d e f)}^{2 / 6} a b + b c + c d + d e + e f + f a \geq 6 {(a b c d e f)}^{1 / 3} (2)$
Dividing $(2)$ by $(1)$
$a b + b c + c d + d e + e f + f a \geq 24$

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