Questions: The graph below shows the function f(x). At which of the labelled points is the derivative f'(x) positive? (a) A and B (b) A and D (c) B and C (d) B and D (e) C and D

The graph below shows the function f(x). At which of the labelled points is the derivative f'(x) positive? (a) A and B (b) A and D (c) B and C (d) B and D (e) C and D
Transcript text: The graph below shows the function $f(x)$. At which of the labelled points is the derivative $f^{\prime}(x)$ positive? (a) A and B (b) A and D (c) B and C (d) B and D (e) C and D
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Solution

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Solution Steps

Step 1: Understand the Problem

We need to determine at which of the labeled points (A, B, C, D) the derivative \( f'(x) \) of the function \( f(x) \) is positive. The derivative \( f'(x) \) is positive where the function \( f(x) \) is increasing.

Step 2: Analyze the Graph
  • Point A: The function is decreasing as it moves towards point A.
  • Point B: The function is increasing as it moves away from point B.
  • Point C: The function is decreasing as it moves away from point C.
  • Point D: The function is decreasing as it moves away from point D.
Step 3: Identify Points with Positive Derivative

From the analysis:

  • At point B, the function is increasing, so \( f'(x) \) is positive.
  • At point C, the function is increasing, so \( f'(x) \) is positive.

Final Answer

The derivative \( f'(x) \) is positive at points B and C.

Answer: (c) B and C

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