Final answer:
The function 1/f(x) is Riemann integrable on the interval (a, b) if f(x) is bounded and does not approach zero at the endpoints, since otherwise 1/f(x) may become unbounded.
Step-by-step explanation:
The question pertains to a function f(x) which is Riemann integrable on a given interval a, b. Because f(x) is positive and bounded on this interval, we know that it does not approach infinity and does not become negative. The main point of interest is whether the reciprocal of this function, 1/f(x), is also Riemann integrable on the same interval.
One important condition for Riemann integrability is that a function must be bounded and not have an infinite number of discontinuities on the interval of interest. Since f(x) is given as Riemann integrable, it is bounded by definition, and so too is 1/f(x) since the reciprocal of a positive number remains finite. However, integrability of 1/f(x) would further depend on the behavior of f(x); for instance, if f(x) approaches zero as x approaches either a or b, then 1/f(x) could become unbounded, which would generally imply that it is not Riemann integrable. Without additional information about the behavior of f(x) near the endpoints a and b, we cannot make a definitive conclusion.
Therefore, the correct answer would be c) Riemann integrable on ( a, b ) with a specific condition.