设实数a,b,c,满足a^2+2b^2+3c^2=3/2,求1/2^a+1/4^b+1/8^c的最小值

设实数a,b,c,满足a^2+2b^2+3c^2=3/2,求1/2^a+1/4^b+1/8^c的最小值

(1/2)^a+(1/4)^b+(1/8)^c
=(1/2)^a+(1/2)^2b+(1/2)^3c
≥3[(1/2)^a•(1/2)^2b•(1/2)^3c]^(1/3)
=3[(1/2)^(a+2b+3c)]^(1/3)
=3(1/2)^(1/2)=3√2/2
即最小值为3√2/2