Answer:
As a result, the correct answer is b.
Explanation:
To determine which statement must be true about the function F, let's analyze the given options:
a. F has a differentiable inverse function H, and H'(f/4) = f(0).
b. None of them.
c. F has a differentiable inverse function H, and H'(0) = (1/2)f(1/4).
d. F does not have an inverse function.
We need to consider the properties and conditions provided in the question.
The function F is defined as F(x) = tan(x) * f(arctan(s)) ds. Here are some important observations:
The range of F is (0, +∞), which means the function takes positive values only.
The given interval for f is (-1/2, 1/2), and the range of F is (0, +∞). This suggests that F is a strictly increasing function.
Based on these observations, we can eliminate options a and d. Option a suggests that F has a differentiable inverse function, but it doesn't specify any conditions related to the properties of F. Option d states that F does not have an inverse function, which is not consistent with the properties of F.
Now let's consider option c. It states that F has a differentiable inverse function H, and H'(0) = (1/2)f(1/4). This option provides specific information about the derivative of the inverse function at a particular point. However, the information given in the question does not provide any direct relation between the values of F and its inverse function. Therefore, we cannot determine the validity of option c based on the given information.
As a result, the correct answer is b. None of the given statements can be determined to be true based on the information provided.