If the graph of f ′(x) is continuous and decreasing with an x-intercept at x = c, which of the following must be true?
The graph of f has a relative maximum at x = c.
The graph of f has an inflection point at x = c.
The graph of f has a relative minimum at x = c.
None of the above is necessarily true.
It should be A
The slope is zero at x = c
If it continues to decrease after that then that must have been an inflection point where it leveled out, then resumed its dive.
Yes :) That would be my pick.
To determine which of the statements must be true, let's analyze the given information.
The fact that the graph of f′(x) is continuous and decreasing means that f′(x) does not have any abrupt changes or jumps in value and that as x increases, the values of f′(x) are getting smaller.
Given that f′(x) has an x-intercept at x = c, it means that f′(c) = 0. This implies that the graph of f has a critical point at x = c.
Now, let's consider each statement and its implications:
1. The graph of f has a relative maximum at x = c:
For f to have a relative maximum at x = c, f′(x) must change from positive to negative as x increases through c. Since f′(x) is continuous and decreasing, it means that for x slightly less than c, f′(x) is positive, and for x slightly greater than c, f′(x) is negative. Therefore, the statement does not necessarily hold true.
2. The graph of f has an inflection point at x = c:
For f to have an inflection point at x = c, the concavity of f must change at that point. This would require f′′(x) (the second derivative of f) to change sign at c. However, no information about f′′(x) is given, so we cannot determine whether an inflection point exists at x = c. Therefore, the statement does not necessarily hold true.
3. The graph of f has a relative minimum at x = c:
For f to have a relative minimum at x = c, f′(x) must change from negative to positive as x increases through c. Since f′(x) is decreasing and it is 0 at x = c, it means that for x slightly less than c, f′(x) is negative, and for x slightly greater than c, f′(x) is positive. Therefore, the statement must be true.
Based on the analysis, we conclude that the only statement that must be true is: "The graph of f has a relative minimum at x = c." Therefore, the correct answer is: The graph of f has a relative minimum at x = c.