证明题1```

令 y = (a + 1/a)(b + 1/b)
则展开得 y = ab + b/a + a/b + 1/ab
= ab + (a^2 + b^2)/ab + 1/ab
= ab + ((a + b)^2 - 2ab)/ab + 1/ab
= ab + 1/ab - 2 + 1/ab
= ab + 2/ab - 2
设 x = ab
则 y = f(x) = x + 2/x - 2
因为a + b = 1 >= 2sqrt(ab) = 2sqrt(x)
所以x <= 1/4
在x <= sqrt(2)的区间上f(x) = x + 2/x - 2递减
所以当x <= 1/4时，
y = f(x) >= f(1/4) = 1/4 + 4 * 2 - 2 = 25/4