the graph of f ′(x) is continuous and decreasing with an x-intercept at x = –3. which of the following statements must be true?
a) the graph of f is always concave up.
b) the graph of f has an inflection point at x = –3.
c) the graph of f has a relative minimum at x = 4.
d) none of these is true.
please help i've been stuck on this question for a while now. i think the answer might be a, but i'm not entirely sure.
It always sheds water, more and more steeply as x increases.
That means always concave DOWN
None of the others are true either.
To determine which statements are true, we need to analyze the given information about the graph of f ′(x). Let's break down each statement and see if it aligns with the given conditions.
a) The graph of f is always concave up.
For a function to be concave up, its derivative must be increasing. However, the given information tells us that f ′(x) is decreasing. Therefore, we can conclude that statement a is NOT true.
b) The graph of f has an inflection point at x = –3.
An inflection point occurs where the concavity of a function changes. Since f ′(x) is continuously decreasing, the concavity should not change. Consequently, statement b is NOT true.
c) The graph of f has a relative minimum at x = 4.
The given information doesn't provide any details about f's behavior at x = 4, so we cannot infer anything about a relative minimum at that point. Therefore, statement c is NOT necessarily true.
By eliminating the previous statements, we are left with only one option:
d) None of these is true.
Hence, the correct answer is d.
To determine which of the statements must be true, let's analyze each statement:
a) The graph of f is always concave up.
Based on the given information that f'(x) is continuous and decreasing, we can determine the concavity of f. Since f'(x) is decreasing, it means that the slope of f(x) is getting smaller as x increases. This corresponds to a concave down shape for f(x), not concave up. Therefore, statement a) is not true.
b) The graph of f has an inflection point at x = -3.
An inflection point occurs when the concavity of a function changes. Since the graph of f'(x) is continuous and decreasing, it means that f''(x) (the second derivative of f) is less than zero, implying that f(x) is concave down. Therefore, there is no change in concavity, and there is no inflection point. Therefore, statement b) is not true.
c) The graph of f has a relative minimum at x = 4.
The information given does not provide any insight into the behavior of f at x = 4. Therefore, statement c) cannot be determined from the given information.
In conclusion, none of the statements a), b), or c) must be true. Therefore, the correct answer is d) none of these is true.