That is √(ab) ≤ 1/(2√2), (1)
a^4/b+32b^4/a≥2√[(a^4/b)(32b^4/a)]= 8√2√(a^3b^3)，
From (1), 8 √ 2 √ (a3b3) ≤ 8 √ 21(16 √ 2) =1/2 ≤ a4/b+32b4/a.
The minimum value of a 4/b+32b 4/a is 1/2, and the minimum value is obtained when a = 1/2 and b = 1/4.