Final answer:
The question deals with determining probabilities from continuous probability functions over specific intervals, including uniform distributions and odd functions. Therefore, the answer is option (c) (−∞, -3) ∪ (1, ∞).
Step-by-step explanation:
The student's question relates to identifying the interval in which a given polynomial function is being analyzed. The subject of the question is probability distributions in the context of calculus-related functions. Specifically, the context involves determining the probability P(0 < x < 12) or P(0 < x < 4) for a continuous probability function within a specified interval.
The functions may be constant or involve transformations that make them odd, affecting the resulting probability. When the function is constant over an interval, as in the case of a uniform distribution, the probability is found by integrating the function's value over the specified interval.
In the case where the function is odd and symmetrical around the x-axis, the integral over the entire real line is zero.
The function f(x) = x⁴/2−2x³−9x²-12x is a polynomial function. To find the interval where the function is positive, we need to determine where the function is greater than 0.
We can do this by analyzing the signs of each term when plugged into the function. By factoring the function, we get f(x) = x(x-3)(x+2)(x+2). We can see that the function is positive when x > 0 and x < -2. Therefore, the answer is option (c) (−∞, -3) ∪ (1, ∞).