设正数$a,b,c$满足$ab+bc+ca=47$,求$(a^2+5)(b^2+5)(c^2+5)$的最小值_____
解:$(a^2+5)(b^2+5)(c^2+5)=
(a^2+5)(5(b+c)^2+(bc-5)^2)ge(sqrt{5}a(b+c)+sqrt{5}(bc-5))^2
=5(ab+bc+ca-5)^2=5*42^2=8820$
当$a=5,b=4,c=3$时取到最小值.
设正数$a,b,c$满足$ab+bc+ca=47$,求$(a^2+5)(b^2+5)(c^2+5)$的最小值_____
解:$(a^2+5)(b^2+5)(c^2+5)=
(a^2+5)(5(b+c)^2+(bc-5)^2)ge(sqrt{5}a(b+c)+sqrt{5}(bc-5))^2
=5(ab+bc+ca-5)^2=5*42^2=8820$
当$a=5,b=4,c=3$时取到最小值.