If a, b, c, d are non-negative real numbers where $a + b + c + d = 1$ , then the maximum value of $a b + b c + c d + a d$ is

\frac{1}{4}

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none of these

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Solution

The correct option is B $\frac{1}{4}$
$∵ a b + b c + c d + a d = (a + c) (b + d)$
$∴$ By the AM-GM inequality, $(a + c) (b + d) \leq {(\frac{a + b + c + d}{2})}^{2} = \frac{1}{4}$
$∴$ The maximum value is $\frac{1}{4}$

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