The cost of a kind of product produced by a company is 2 yuan, the selling price is 3 yuan, and the annual sales volume is 100000 pieces. In order to obtain better benefits, the company is prepared to put out certain funds for advertising. According to experience, when the annual advertising cost is x (unit: 100000 yuan), the annual sales volume of the product will be y times of the original sales volume, and Y is a quadratic function of X. their relationship is shown in the table: X (100000 yuan) Yuan) 012 y11.51.8… (I) find the functional relationship between Y and X; (II) if profit is regarded as total sales minus cost and advertising expenses, try to write the functional relationship between annual profit s (100000 yuan) and advertising expenses x (100000 yuan); (III) if the annual advertising expenses are x, X ∈ [100000, 300000] yuan, ask what range of advertising expenses is, and the annual profit of the company increases with the increase of advertising expenses enlarge?

The cost of a kind of product produced by a company is 2 yuan, the selling price is 3 yuan, and the annual sales volume is 100000 pieces. In order to obtain better benefits, the company is prepared to put out certain funds for advertising. According to experience, when the annual advertising cost is x (unit: 100000 yuan), the annual sales volume of the product will be y times of the original sales volume, and Y is a quadratic function of X. their relationship is shown in the table: X (100000 yuan) Yuan) 012 y11.51.8… (I) find the functional relationship between Y and X; (II) if profit is regarded as total sales minus cost and advertising expenses, try to write the functional relationship between annual profit s (100000 yuan) and advertising expenses x (100000 yuan); (III) if the annual advertising expenses are x, X ∈ [100000, 300000] yuan, ask what range of advertising expenses is, and the annual profit of the company increases with the increase of advertising expenses enlarge?

(1) Let the analytic expression of the quadratic function be y = AX2 + BX + C. from the relational table, we get C = 1A + B + C = 1.54a + 2B + C = 1.8, and the solution is a = - 110b = 35C = 1. The analytic expression of the function is y = - 110x2 + 35x + 1