The graph of f ′(x) is continuous and decreasing with an x-intercept at x = 0. Which of the following statements is false? (4 points)
The graph of f has an inflection point at x = 0.
The graph of f has a relative maximum at x = 0.
The graph of f is always concave down.
The graph of the second derivative is always negative.
I know that the first two are right, but not sure about the second two.
The graph of f has an inflection point at x = 0. is the answer
The graph of f has an inflection point at x = 0 - This statement is true. If the first derivative (f'(x)) is continuous and changing from positive to negative (decreasing), then it indicates a change in concavity and presence of an inflection point.
The graph of f has a relative maximum at x = 0 - This statement is also true. Since f'(x) is decreasing and changes from positive to negative at x = 0, it implies that f(x) has a local maximum at x = 0.
The graph of f is always concave down - This statement is false. Since f'(x) is decreasing, it means that the slope of f(x) is becoming less steep as x increases. However, it does not necessarily imply that the graph of f(x) is always concave down. It could still have regions of concave up.
The graph of the second derivative is always negative - This statement is also false. The second derivative of f(x) represents the concavity of the graph. Since f'(x) is continuous and decreasing, it can be inferred that the second derivative is negative for x < 0 and positive for x > 0. The graph of the second derivative changes from negative to positive at x = 0, indicating a change in concavity.
To determine which of the following statements is false, we need to consider the properties of a function and its derivatives.
The given information states that the graph of f ′(x) is continuous and decreasing with an x-intercept at x = 0. Let's analyze each statement to find the false one:
1. The graph of f has an inflection point at x = 0.
To determine if this statement is true, we need more information about the behavior of f ″(x), the second derivative. An inflection point occurs when the graph changes concavity, meaning the sign of the second derivative changes. Since we don't have information about f ″(x), we can't conclude whether this statement is true or false confidently.
2. The graph of f has a relative maximum at x = 0.
Since f ′(x) is decreasing and changes sign from positive to negative at x = 0, this implies that the function f(x) is also changing from increasing to decreasing at x = 0. Therefore, statement 2 is true.
3. The graph of f is always concave down.
The concavity of the graph depends on the sign of f ″(x), the second derivative. Without knowing the behavior of f ″(x), we cannot determine whether the graph is always concave down. Thus, we cannot conclude whether this statement is true or false.
4. The graph of the second derivative is always negative.
The given information only tells us that f ′(x) is decreasing, which means the slope of the graph of f is decreasing. However, it doesn't provide direct information about the sign of f ″(x). We cannot conclude whether the second derivative is always negative based on the given information. Hence, we can't determine if this statement is true or false.
To summarize, statements 1 and 4 cannot be determined based on the given information. Statements 2 is true, and it is unclear whether statement 3 is true or false. Therefore, statement 3 is the false statement.
since f' is decreasing, we know that f" < 0
So, at x=0, the slope is 0, and the curve is concave down, so it is a max there.
if f" is always negative, the curve is always concave down.
Looks like they're all true.
Consider the parabola y = -x^2
f' = -2x, which is always decreasing, and f'(0) = 0
f" = -2, and the parabola is always concave down.
So, only #1 is false.