The maximum value of the expression, $f (x, y, z) = x y z (4 d - a x - b y - c z)$ where a, b, c, d are positive constants, x, y, z are positive variables and $4 d - a x - b y - c z > 0$ is :

\frac{a b c}{d}

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\frac{d^{2}}{a b c}

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\frac{a b c}{d^{3}}

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\frac{d^{4}}{a b c}

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Solution

The correct option is D $\frac{d^{4}}{a b c}$
Using $A . M \geq G . M$
$\frac{(a x) + (b y) + (c z) + (4 d - a x - b y - c z)}{4} \geq [a x . b y . c z . (4 d - a x - b y - c z)]^{1 / 4}$
$\Rightarrow a b c x y z (4 d - a x - b y - c z) \leq d^{4}$
$\Rightarrow x y z (4 d - a x - b y - c z) \leq \frac{d^{4}}{a b c}$

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