Question

若a、b都为正数,且1/a+1/b=1,则（2+b)/2ab的最大值为

Answer 1

1/a+1/b=1

(a+b) =ab

a(b-1) =b

a= b/(b-1)

S = (2+b)/(2ab)

= (2+b)(b-1)/( 2b^2)

S' = [2b^2(2b+1) - 4b(b^2+b-2)] /(4b^4)

S' =0

b(2b+1)-2(b^2+b-2) =0

-b+4=0

b=4

S''(4) a=4/3

max S = (2+4)/(2(4/3)4) = 9/16