Is the product of four consecutive even integers positive?
Let the four consecutive even integers be 2n-2, 2n , 2n+2, 2n+4, where n is an integer.

1. The sum of these integers is positive but smaller than 20

Sun of these integers = 2n- 2 + 2n + 2n+2 + 2n+4 = 8n+4.
0<8n+4<20
-4<8n<16.
-1/2<n<2. Since, n is an integer, it can be 0 or 1.
If n = 0, the integer set is -2,0,2,4. The product of the members of the set is 0.
If n = 1, the integer set is 0,2,4,6. The product of the members of the set is 0.
For both the cases, product of four consecutive even integers is NOT positive. So, Statement I is sufficient to answer the question.

2. The product of the smallest two of these integers is positive

Case 1: If the set is -4,-2,0,2, then the product of these integers is 0.
Case 2: If the set is 2,4,6,8, then the product of these integers is greater than 0.
Since we don't have a definite answer, Statement II is Insufficient to answer the question.

IMO A

Do you also know that the product of four consecutive integers is either zero or positive.

Case 1: 1,2,3,4 - The product is Positive.
Case 2: 0,1,2,3 - The product is 0.
Case 3: -3,-2,-1,0 - The product is 0.
Case 3: -4,-3,-2,-1 - The product is Positive.

Do you also know that the product of four consecutive even integers is either zero or positive.

Case 1: 2,4,6,8 - The product is Positive.
Case 2: 0,2,4,6 - The product is 0.
Case 3: -6,-4,-2,0 - The product is 0.
Case 3: -8,-6,-4,-2 - The product is Positive.