If $a,b,c,d$ are positive real numbers and $abcd=1$ then Find the minimum value of $(4+a)(4+b)(4+c)(4+d)$. Find the condition when minimium value holds.
I've used AM-GM Inequality $4+a \ge 2 \sqrt{4a}$, $4+b \ge 2 \sqrt{4b}$, $4+c \ge 2 \sqrt{4c}$, $4+d \ge 2 \sqrt{4d}$. Since $a,b,c,d$ are positive we can multiply these inequalities... $(4+a)(4+b)(4+c)(4+d) \ge 256$. I stuck at when finding the condition for equality holds.