Various values for the derivative, f'(x), of a differentiable function f are shown below.
x=1, f'(x) = 8
x=2, f'(x) = 4
x=3, f'(x) = 0
x=4, f'(x) = -4
x=5, f'(x) = -8
x=6, f'(x) = -12
If f'(x) always decreases, then which of the following statements must be true?
A) f(x) has a relative minimum at x=3
B) f(x) is concaved upward for all x
C) The graph of f(x) is symmetric with the line x=3
D) f(x) changes concavity at x=3
E) f(x) has a relative maximum at x=3
It has to be A or E. You have relative maxima or minima (or an inflection point) when the derivative is zero
(A) can be ruled out because the second derivative is negative at x = 3.
What choice does that leave you with?
To determine which statement is true given that f'(x) always decreases, we need to use the concepts of the first and second derivative tests.
The first derivative, f'(x), represents the rate of change of the function f(x) at a particular point. If f'(x) is decreasing, it means that the function's rate of change is becoming smaller as x increases.
Now, let's analyze each statement:
A) f(x) has a relative minimum at x=3:
To determine the presence of a relative minimum, we need to check the sign changes in f'(x). But since f'(x) always decreases, it means that it is always negative or zero. Therefore, there are no sign changes, and f(x) cannot have a relative minimum at x=3. So, statement A is not necessarily true.
B) f(x) is concaved upward for all x:
The concavity of a function is determined by the sign of the second derivative, f''(x). Since we only have information about f'(x) and not f''(x), we cannot determine the concavity of f(x). So, statement B cannot be concluded.
C) The graph of f(x) is symmetric with the line x=3:
Symmetry in a graph usually indicates that the function has even or odd symmetry. However, without additional information, we cannot determine the symmetry of the graph based solely on the values of f'(x). So, statement C cannot be concluded.
D) f(x) changes concavity at x=3:
To determine if f(x) changes concavity at x=3, we need to examine the behavior of the second derivative, f''(x). Unfortunately, we don't have any information about f''(x). So, statement D cannot be concluded.
E) f(x) has a relative maximum at x=3:
Similarly to statement A, the presence of a relative maximum depends on the sign changes in f'(x). Since f'(x) always decreases and is negative or zero, there are no sign changes. Therefore, f(x) cannot have a relative maximum at x=3. Thus, statement E is not necessarily true.
In conclusion, the only statement that can be concluded based on the information provided is that none of the statements A, B, C, D, or E can be determined to be true based solely on the given values of f'(x).