Question

设m、n、p为正数,且m^2+n^2-p^2=0,求p\m+n的最小值。

设m、n、p为正数,且m^2+n^2-p^2=0,求p\m+n的最小值。要详细过程哦！

Answer 1

m^2+n^2-p^2=0
m^2+n^2=p^2=(m+n)^2-2mn
(m+n)^2/p^2=1+2mn/p^2<=1+(m^2+n^2)/p^2=2, (m=n时取等号)
(m+n)/p<=根号2
p/(m+n)>=根号2/2

p/(m+n)的最小值是根号2/2。