Question

Let f be a differentiable function with a domain of (0,10). It is known that f′(x), the derivative of f(x), is negative on the intervals (0,2) and (4,6) and positive on the intervals (2,4) and (6,10). Which of the following statements is true?

A) f has no relative minima and three relative maxima.
B) f has one relative minimum and two relative maxima.
C) f has two relative minima and one relative maximum.
D) f has three relative minima and no relative maxima.

Answer 1

The function f(x) has one relative minimum and two relative maxima.

Step-by-step explanation:

The given information tells us that the derivative of the function is negative on the intervals (0,2) and (4,6) and positive on the intervals (2,4) and (6,10). Since f'(x) is negative on the intervals (0,2) and positive on the intervals (2,4), this means that f(x) is decreasing on the interval (0,2) and increasing on the interval (2,4). Similarly, f(x) is decreasing on the interval (4,6) and increasing on the interval (6,10).

Based on the behavior of f'(x) and f(x), we can conclude that f(x) has one relative minimum, which occurs at x = 2, and two relative maxima, which occur at x = 4 and x = 6. Therefore, the correct statement is B) f has one relative minimum and two relative maxima.