If the functions f and g are defined for all real numbers and f is an antiderivative 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 clarification as to why that's the answer. Thank you.
just because the graph is a horizontal line, it is not necessarily the line
y = 0
The statement "If g(x) = 0 for all x, then f(x) = 0 for all x" is NOT necessarily true.
This is because if g(x) = 0 for all x, it means that g(x) is a constant function equal to 0. However, the function f(x) can be any constant function, not just the constant function equal to 0.
For example, let's say g(x) = 0 for all x. Then, any constant function f(x) = C (where C is any constant) will be an antiderivative of g(x) since the derivative of a constant function is 0.
So, in general, if g(x) = 0 for all x, it does not mean that f(x) = 0 for all x. Therefore, option (D) is not necessarily true.
To answer this question, we need to understand the relationship between a function and its antiderivative.
A function f is said to be an antiderivative of g if the derivative of f is equal to g. In other words, f'(x) = g(x) for all x.
Now let's analyze each statement:
(A) If g(x) > 0 for all x, then f is increasing:
This statement is true because if g(x) is positive for all x, it means that the derivative of f, which is g(x), is positive. If the derivative is positive, it implies that the function f is increasing.
(B) If g(a) = 0, then f(x) has a horizontal tangent at x = a:
This statement is also true. If g(a) = 0, it means that at x = a, the derivative of f, which is g(x), is equal to zero. This implies that f(x) has a horizontal tangent at x = a.
(C) If f(x) = 0 for all x, then g(x) = 0 for all x:
This statement is true because if f(x) is identically zero for all x, it means that the derivative of f, which is g(x), is also equal to zero for all x.
(D) If g(x) = 0 for all x, then f(x) = 0 for all x:
This statement is NOT necessarily true. Just because g(x) is zero for all x does not imply that the antiderivative f(x) will be zero for all x. There could be a constant of integration present in f(x) that allows it to take non-zero values even when g(x) is zero.
(E) f is continuous for all x:
This statement is true. When f is an antiderivative of g, it follows that f(x) is continuous for all x in the domain of g.
Therefore, the correct answer is (D) If g(x) = 0 for all x, then f(x) = 0 for all x.