Just to illustrate with the case of n = 2. Then we have

(2 + a)(2 + 1/a) = 5 + 2(a + 1/a)

But a + 1/a >= 2 for all positive real a, so f(2) >= 9.

[If a + 1/a < 2 and a>0, then a^2 -2a +1 < 0, (a - 1)^2 < 0, which isn't possible.]