Let a, B ∈ positive real number, and a + B = 1, the proof is greater than or equal to 25 / 4

Let a, B ∈ positive real number, and a + B = 1, the proof is greater than or equal to 25 / 4

It should be (a + 1 / a) (B + 1 / b) ≥ 25 / 4
（a+1/a)(b+1/b)
=（a²+1)(b²+1)/(ab)
=(a²+b²+a²b²+1)/(ab)
=[(a²+b²+2ab)-2ab+a²b²+1]/(ab)
=[(a+b)²+a²b²-2ab+1]/(ab) 【a+b=1]
=(a²b²-2ab+2)/(ab)
=ab+2/(ab)-2
∵a+b=1,a>0,b>0
a+b≥2√(ab)
∴ab≤1/4
The function AB + 2 / (AB) is a decreasing function
When AB = 1 / 4, the minimum value of AB + 2 / (AB) is 1 / 4 + 8 = 33 / 4
∴ab+2/(ab)-2≥33/4-2=25/4
That is (a + 1 / a) (B + 1 / b) ≥ 25 / 4