Question

Determine the interval(s) on which the given function is decreasing.

A. (−[infinity],−1)∪(0,[infinity])
B. (1,[infinity])
C. (−[infinity],−1)∪(1,[infinity])
D. (−1,0)

Answer 1

The function f(x) is decreasing over the intervals:
`$(\infty,-1)\cup(0,\infty)$` and the correct option is A.

To determine the interval(s) on which the given function is decreasing, we can analyze the derivative of the function. If the derivative is negative for a certain interval, it indicates that the function is decreasing over that interval.

Step 1: Differentiate the function

f(x) = x^3 - 3x^2 - x + 1

F(x) = 3x^2 - 6x - 1

Step 2: Set the derivative to zero and solve for x

F(x) = 0

3x^2 - 6x - 1 = 0

(x - 1)(3x + 1) = 0

x = 1 or x = -1/3

Step 3: Analyze the signs of the derivative in different intervals

Interval | x-value | f'(x) | Verdict

x < -1/3 | x = -1 | f'(-1) = -1 < 0 | f(x) is decreasing

-1/3 < x < 1 | x = 0 | f'(0) = -1 < 0 | f(x) is decreasing

x > 1 | x = 2 | f'(2) = 5 > 0 | f(x) is increasing

Step 4: Identify decreasing intervals

Based on the analysis, the function f(x) is decreasing over the intervals:

A. (−[∞],−1)∪(0,[∞])