The graph of f ′(x) is continuous and decreasing with an x-intercept at x = 2. Which of the following statements must be true?
Edit
The graph of f has an inflection point at x = 2.
The graph of f has a relative maximum at x = 2.
The graph of f is always concave down. MY ANSWER*******
The graph of f is always increasing
I agree.
Thank you!!
The correct answer is: The graph of f is always concave down.
To determine which of the statements must be true, we need to analyze the given information.
We are told that the graph of f'(x) is continuous and decreasing with an x-intercept at x = 2.
Let's break it down:
1. The fact that f'(x) is continuous means that there are no sudden changes or breaks in the slope of f(x). The graph of f(x) can have sloping lines, but there are no sharp turns or jumps.
2. The fact that f'(x) is decreasing means that the slope of f(x) is getting smaller as x increases. This indicates that the graph of f(x) is becoming less steep or flatter as x increases.
3. The x-intercept at x = 2 means that the graph of f'(x) crosses the x-axis at x = 2. This suggests that at x = 2, the slope of f(x) is 0, which can be an indicator of an extremum point.
Now let's analyze the given statements:
1. The statement "The graph of f has an inflection point at x = 2" cannot be determined based on the given information. An inflection point occurs where the concavity of a graph changes. We don't have information about the concavity of f(x), so we cannot conclude if there is an inflection point at x = 2.
2. The statement "The graph of f has a relative maximum at x = 2" cannot be determined based on the given information. A relative maximum occurs when the graph reaches a peak value. The decreasing nature of f'(x) does not provide enough information to determine if a relative maximum occurs at x = 2.
3. The statement "The graph of f is always concave down" is the answer. Since f'(x) is decreasing, it implies that the slope of f(x) is always negative or decreasing. This indicates that the graph of f(x) is always concave down, meaning it curves downward like a smile.
Therefore, the correct answer is "The graph of f is always concave down."