Final answer:
The smallest positive integer 'a' such that the Intermediate Value Theorem guarantees a zero exists between 0 and 'a' for the given polynomial function is 3.
Step-by-step explanation:
The polynomial function is given as f(x) = 2x2 - 2x - 3. We need to determine the smallest positive integer 'a' such that the Intermediate Value Theorem guarantees a zero exists between 0 and 'a'.
To apply the Intermediate Value Theorem, we can evaluate the function at two different points: f(0) and f(a), and check if the function values have opposite signs. Since f(0) = -3 and f(a) = 2a2 - 2a - 3, we need to find a value of 'a' for which f(0) and f(a) have opposite signs. Starting with 'a' = 1, we find f(0) = -3 and f(1) = -3. Since the function values are the same, we increment 'a' by 1.
By evaluating f(0), f(1), f(2), f(3), and f(4), we find that f(3) = 9 and f(4) = 19. The function values differ in sign, indicating that a zero exists between 0 and 3.