Question

The function f has a first derivative given by f'(2) = x(x - 3)(x + 1). At what values of x does f have a relative maximum?

A) -1 only

B) 0 only

C) -1 and 0 only

D) -1 and 3 only

E) -1, 0, and 3

Answer 1

The function f has relative maxima at x = -1 and x = 3.(Option E)

Step-by-step explanation:

To find the values of x where the function f has relative maxima, we need to analyze the behavior of the first derivative f'(x). The critical points occur where f'(x) = 0 or is undefined. In this case, f'(2) is given as x(x - 3)(x + 1).

1. **Finding Critical Points:** Set f'(x) equal to zero and solve for x:

The solutions are x = -1, 0, and 3.

2. Determining Relative Extrema: Evaluate the second derivative test or examine the sign changes in the intervals determined by the critical points. If f''(x) is positive before x = -1 and negative after, it indicates a relative maximum at x = -1. Similarly, if f''(x) changes from negative to positive at x = 3, it suggests a relative maximum at x = 3.

3. Conclusion: The critical points are x = -1, 0, and 3. After analyzing the behavior of the first derivative, we find that f has a relative maximum at x = -1 and x = 3. Therefore, the correct answer is E) -1, 0, and 3. (Option E)