Final answer:
The Intermediate Value Theorem guarantees that a continuous function on a closed interval which begins at 3 and ends at 6 will take on every value between 3 and 6. Therefore, the function will reach 4 and 5, but not necessarily 2 or 7.
Step-by-step explanation:
The Intermediate Value Theorem (IVT) tells us that if f is a continuous function on a closed interval and f(a) and f(b) are the values at the endpoints of the interval, then f takes on every value between f(a) and f(b) at some point within that interval. Given that in this question the function is continuous on the interval from -2 to 1 and takes on values 3 at f(-2) and 6 at f(1), the IVT ensures that f(x) will take on all values between 3 and 6. Therefore, the statements A. f(x) = 4 for some x in -2, 1 and B. f(x) = 5 for some x in -2, 1 are guaranteed by the IVT. However, the IVT does not guarantee that f(x) will take on values outside the range of 3 to 6, such as 2 or 7, so statements C and D are not necessarily true.