Question

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

• A. $32$
• B. $24$
• C. $56$
• D. $48$

Answer 1

The correct option is A

$32$

$a^7{b}^2{c}^3{d}^5=512=2^9$
$AM\ge G.M$
$\frac{a^7+2b^2+2c^3+2d^5}{4}\ge \sqrt[4]{4a^7\cdot 2b^2\cdot 2c^3\cdot 2d^5}$
$\frac{a^7+2b^2+2c^3+2d^5}{4}\ge \sqrt[4]{42^{12}}$
$a^7\cdot 2b^2\cdot 2c^3\cdot 2d^5\ge 8.4$
$\ge 32$