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Real Valued Functions

a7b2c3d5=512....

Question

$a^{7} b^{2} c^{3} d^{5} = 512.$ Find the minimum value of $a^{7} + 2 b^{2} + 2 c^{3} + 2 d^{5}$

A

$32$

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B

$24$

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C

$56$

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D

$48$

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Solution

The correct option is A
$32$

$a^{7} b^{2} c^{3} d^{5} = 512 = 2^{9}$
$A M \geq G . M$
$\frac{a^{7} + 2 b^{2} + 2 c^{3} + 2 d^{5}}{4} \geq \sqrt[4]{a^{7} \cdot 2 b^{2} \cdot 2 c^{3} \cdot 2 d^{5}}$
$\frac{a^{7} + 2 b^{2} + 2 c^{3} + 2 d^{5}}{4} \geq \sqrt[4]{2^{12}}$
$a^{7} \cdot 2 b^{2} \cdot 2 c^{3} \cdot 2 d^{5} \geq 8.4$
$\geq 32$

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