If the graph of f " (x) is continuous and has a relative maximum at x = c, which of the following must be true?
The graph of f has an x-intercept 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.
The Answer is D
To determine which statement must be true, we need to understand the properties of the derivative function, f'(x), and how they relate to the original function, f(x).
Given that the graph of f" (x) (the second derivative) is continuous and has a relative maximum at x = c, we can make the following observations:
1. If the graph of f'(x) has a relative maximum at x = c, it means that f'(c) = 0 and f''(c) < 0. This indicates that at x = c, the slope of the tangent line is horizontal (f'(c) = 0), and the curvature of the graph is concave downward (f''(c) < 0).
2. Since f'(x) represents the slope of f(x), if f'(c) = 0, it implies that the graph of f(x) has a horizontal tangent line at x = c. However, a horizontal tangent line does not guarantee an x-intercept at x = c. Therefore, the statement "The graph of f has an x-intercept at x = c" is not necessarily true.
3. An inflection point occurs where the concavity of the graph changes. Since f''(c) < 0, indicating concave downward curvature at x = c, it means that there is no change in concavity around x = c. Therefore, the statement "The graph of f has an inflection point at x = c" is also not necessarily true.
4. Since f'(x) has a relative maximum at x = c, it means that the slope of f(x) is changing from positive to negative as x approaches c. This suggests that f(x) has a decreasing behavior on either side of x = c. Considering this, it is likely that f(x) has a relative minimum at x = c. Thus, the statement "The graph of f has a relative minimum at x = c" is a possible conclusion from the given information.
Based on the analysis above, the correct answer is: The graph of f has a relative minimum at x = c.