Question

若实数a,b满足2的a次方+2的b次方=2的a+b次方,则2的a的次方+2的b+1次方的最小值

Answer 1

x=2^a,y=2^b
条件等价于
x+y=xy
所求式子等价于x+2y
x+y=xy得到
(x-1)(y-1)=1
x+2y = (x-1) + 2(y-1) + 3
≥2根号[ 2(x-1)(y-1) ] +3
=2√2 + 3
当x=1+√2,y=1+√2/2时取得