已知两正数x,y满足x+y=1,求证：(x+1/x)^2+(y+1/y)^2>=25/2
1个回答

as x + y = 1,we have 2*sqrt(xy) = 1/(1/4)^2 = 16 ----------- (2)

then we have original expression:

x^2 + 2 + 1/x^2 + y^2 + 2 + 1/y^2

= x^2 + y^2 + (x^2 + y^2)/(xy)^2 + 4

as (1) and (2),the expression above can be rewritten as

.

>= 1/2 + 16/2 + 4 = 25/2

so the required inequality is achieved and equality is valid when and only when x = y = 1/2

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