Question

In which interval is the function f(x) = x cubed divided by 3 - x 3 increasing?

1) (-[infinity], -1) only
2) (1, [infinity]) only
3) (-[infinity], -1) u (1, [infinity])
4) (-1, 1) only

Answer 1

The function f(x) = x cubed divided by 3 - x is increasing in the intervals (-∞, -1) and (1, ∞), found by analyzing the positive regions of its derivative, f'(x) = x^2 - 1.

Step-by-step explanation:

To determine in which interval the function f(x) = x3 / 3 - x is increasing, we look at its derivative, f'(x). The function increases where its derivative is positive. To find the critical points, set the derivative equal to zero:

f'(x) = d/dx (x3/3) - d/dx (x) = x2 - 1

The derivative equals zero when x2 - 1 = 0, which is when x = -1 or x = 1. Thus, we test intervals around these points: (-∞, -1), (-1, 1), and (1, ∞).

In the intervals (-∞, -1) and (1, ∞), f'(x) is positive, meaning the function is increasing. In the interval (-1, 1), f'(x) is negative, and the function is decreasing. Therefore, the correct answer is option 3) (-∞, -1) ∪ (1, ∞), indicating that the function is increasing in the intervals (-∞, -1) and (1, ∞).