Question

Et and be inverse functions that are differentiable for all x. If f'(x) = g(x) and g'(x) = eˣ, which of the following statements must be false?

A. f(x) and g(x) are not inverse functions.
B. The derivative of f(x) is eˣ.
C. The derivative of g(x) is eˣ.
D. f(x) and g(x) are not differentiable.

Answer 1

To determine which of the given statements must be false, we analyze the information provided about the functions f(x) and g(x) and their derivatives. The false statement is A. f(x) and g(x) are not inverse functions.

Step-by-step explanation:

To determine which of the given statements must be false, we need to analyze the information provided. We are given that the functions f(x) and g(x) are inverse functions and differentiable for all x. We are also given that f'(x) = g(x) and g'(x) = e^x. Let's analyze each statement:

A. f(x) and g(x) are not inverse functions: This statement contradicts the given information, so it must be false.

B. The derivative of f(x) is e^x: Since f'(x) = g(x), and g'(x) = e^x, this statement is true.

C. The derivative of g(x) is e^x: This statement is consistent with the given information, so it is true.

D. f(x) and g(x) are not differentiable: This statement contradicts the given information, so it must be false.

Based on our analysis, the false statement is A. f(x) and g(x) are not inverse functions.