If the functions f and g are defined for all real numbers and f is an antiderative of g, which of the following statements is NOT necessarily true?
(A)If g(x)>0 for all x, then f is increasing
(B)If g(a)=0, then f(x) has a horizontal tangent at x=a
(C)If f(x)=0 for all x, then g(x)=0 for all x
(D)If g(x)=0 for all x, then f(x)=0 for all x
(E)f is continuous for all x
I know the answer is D, but I need an explanation as to why, please and thanks.
Just because the graph is constant, that does not mean that f = 0.
F(6)=
In this scenario, if f is an antiderivative of g, then we can say that the derivative of f is equal to g. That is, f'(x) = g(x).
Now, let's analyze each statement:
(A) If g(x) > 0 for all x, then f is increasing:
Since the derivative of f is equal to g, if g(x) > 0, it implies that f'(x) > 0, which means that f is indeed increasing. Therefore, this statement is necessarily true.
(B) If g(a) = 0, then f(x) has a horizontal tangent at x = a:
If g(a) = 0, it means that f'(a) = 0 (since f'(x) = g(x)). And if f'(a) = 0, it implies that f(x) has a horizontal tangent at x = a. So, this statement is also necessarily true.
(C) If f(x) = 0 for all x, then g(x) = 0 for all x:
Since f(x) = 0 for all x, it means that f'(x) = 0 for all x (as f'(x) = g(x)). However, this does not necessarily mean that g(x) = 0 for all x. g(x) could have non-zero values at some points and still have a derivative equal to zero at those points. Therefore, this statement is not necessarily true.
(D) If g(x) = 0 for all x, then f(x) = 0 for all x:
Since f'(x) = g(x), if g(x) = 0 for all x, it implies that f'(x) = 0 for all x. However, this does not necessarily mean that f(x) = 0 for all x. The antiderivative of a function with value 0 at all points can still have non-zero values at some points (like a constant value between any two zeros of g(x)). So, this statement is not necessarily true.
(E) f is continuous for all x:
Since f'(x) = g(x), and g(x) is assumed to be defined for all real numbers, it implies that the derivative f'(x) also exists for all real numbers. Therefore, f is differentiable for all x, which implies that f is continuous for all x. Hence, this statement is necessarily true.
In summary, the statement that is NOT necessarily true is (D) If g(x) = 0 for all x, then f(x) = 0 for all x.
To determine which statement is NOT necessarily true, we need to analyze each statement and understand whether it holds in all cases.
(A) If g(x) > 0 for all x, then f is increasing:
This statement is true. If the derivative of a function g(x) is always positive (i.e., g(x) > 0), then the function f(x), which is the antiderivative of g(x), will be strictly increasing. Therefore, statement (A) is necessarily true.
(B) If g(a) = 0, then f(x) has a horizontal tangent at x = a:
This statement is true. If g(a) = 0, it means that at point x = a, the function g(x) crosses the x-axis. Since the antiderivative f(x) accumulates the area under the curve of g(x), when g(x) crosses the x-axis, f(x) will have a horizontal tangent. Therefore, statement (B) is necessarily true.
(C) If f(x) = 0 for all x, then g(x) = 0 for all x:
This statement is true. If the antiderivative f(x) is equal to 0 for all x, it means that the area under the curve of g(x) is 0 for all x. This implies that g(x) is equal to 0 for all x. Therefore, statement (C) is necessarily true.
(D) If g(x) = 0 for all x, then f(x) = 0 for all x:
This statement is NOT necessarily true. If g(x) is always 0 for all x, it means that the area under the curve of g(x) is always 0. However, this does not necessarily imply that the antiderivative f(x) must be 0 for all x. The constant value of f(x) can be any real number. Hence, statement (D) is NOT necessarily true and is the correct answer.
(E) f is continuous for all x:
This statement is true. Since f(x) is an antiderivative of g(x), which is defined for all real numbers, f(x) must also be defined for all real numbers. Therefore, f(x) is continuous for all x. Thus, statement (E) is necessarily true.
In summary, the statement that is NOT necessarily true is (D) If g(x) = 0 for all x, then f(x) = 0 for all x.