Final answer:
To find where the function f(x) = 3x^3 - 9x - 8 is increasing, we calculate its derivative, determine the critical points, and use the first derivative test. The function is increasing on the intervals (-∞, -1) and (1, ∞), making option (c) the closest to the correct answer despite a typo.
Step-by-step explanation:
To find all intervals where the function f(x) = 3x^3 - 9x - 8 is increasing, one must first determine the function's critical points. This involves finding the derivative of the function, f'(x), and solving for where this derivative is equal to zero or is undefined, as these points represent potential maxima, minima, or points of inflection.
The first derivative of f(x) is f'(x) = 9x^2 - 9. Setting this equal to zero, we get 9x^2 - 9 = 0 which simplifies to x^2 = 1. This gives us critical points of x = -1 and x = 1.
Next, we use the first derivative test to determine whether these critical points correspond to intervals of increase or decrease. We test the intervals around our critical points: (-∞, -1), (-1, 1), and (1, ∞). We find that the derivative is positive (indicating increase) for x < -1 and x > 1, while it is negative (indicating decrease) for -1 < x < 1.
Therefore, the intervals where the function f(x) is increasing are (-∞, -1) and (1, ∞). Thus, the correct answer is (c) Intervals: (-∞, -∞), (-∞, -1), (1, ∞). It is important to note the typo in this option, as it incorrectly repeats the interval with infinity. Nevertheless, it is the option closest to the correct answer.