The tourism department of a city has developed a tourist souvenir. The cost of each product is 15 yuan, and the price is 20 yuan. Sell one piece a month on average. By improving the technology,

(1) After the process improvement, the price of each product is 20( 1+x), and the average monthly sales volume is a( 1-x2).

So the monthly average profit y=a( 1-x2)? [20( 1+x)- 15]，

The functional relationship between y and x is y = 5a (1+4x-x2-4x3).

Therefore, the functional relationship is: y = 5a (1+4x-x2-4x3) (0 < x <1).

(II) x = 12 or x = 0 from y'=5a(4-2x- 12x2)=0? 23 (house)

Y' > 0； when 0 < x < 12; When 12 < x < 1,

∴ the function y = 5a (1+4x-x2-4x3) (0 < x <1) gets the maximum value at x = 12.

Therefore, after improving the technology, when the sales price of products is 20( 1+ 12)=30 yuan, the average monthly profit of souvenirs sold by the tourism department is the largest.