Final answer:
The function h(x) simplifies to -x, and its absolute minimum value over the interval [-3, 2] is -2, which occurs at x = 2. This corresponds to option (b) h(2).
Step-by-step explanation:
The student is asking to find the absolute minimum value of the function h(x) = -2x² / (2x) over the interval [-3, 2]. To simplify the function, we can cancel out the common factor of '2x' from the numerator and the denominator, resulting in h(x) = -x. This is a linear function, and it is continuous and differentiable over the interval in question. To find the absolute minimum value, we need to evaluate the function at the endpoints of the interval and any critical points within the interval.
However, there is a point to be considered at x=0, where the original function h(x) = -2x² / (2x) is undefined. Thus, we do not consider h(0) in our list of values because it does not exist.
Now let's evaluate:
h(-3) = -(-3) = 3
h(2) = -(2) = -2
h(-1) = -(-1) = 1
Looking at the values, h(2) yields the smallest value, which is -2. Therefore, the absolute minimum value of h(x) over the given interval is -2, which corresponds to option (b) h(2).