Question

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

Answer 1

The function f(x) has two relative minima at x=2 and x=6 where the derivative changes from negative to positive, and one relative maximum at x=4 where the derivative changes from positive to negative. Thus, the correct answer is C.

Step-by-step explanation:

When analyzing the behavior of a differentiable function's graph, given the information about its derivative, we can determine the locations of relative minima and maxima. The function f(x) has a derivative f'(x) that is negative on intervals (0, 2) and (4, 6) and positive on (2, 4) and (6, 10).

A derivative that changes from negative to positive indicates a relative minimum, whereas a derivative that changes from positive to negative indicates a relative maximum.

In this case, at x=2 and x=6, the derivative changes from negative to positive, suggesting that f has relative minima at these points. At x=4, the derivative changes from positive to negative, indicating a relative maximum there. Therefore, f has two relative minima and one relative maximum.

The given information states that ƒ'(x), the derivative of ƒ(x), is negative on the intervals (0, 2) and (4, 6) and positive on the intervals (2, 4) and (6, 10). Based on this, we can determine the relative extrema of the function.

Hence, the correct statement is C: f has two relative minima and one relative maximum.