The discussion revolves around the application of the Inverse Function Theorem in relation to a specific function, particularly focusing on the behavior of its derivative within a given interval. Participants are examining the implications of the theorem when the derivative is zero at certain points.
The discussion is active, with participants presenting differing interpretations of the theorem and its conditions. Some have offered clarifications regarding the necessity of the derivative being nonzero near the point of interest, while others are exploring the implications of choosing different intervals for analysis.
There is a focus on the interval in which the theorem is being applied, with some participants suggesting alternative intervals that avoid points where the derivative is zero. The original poster and others are grappling with the definitions and conditions set forth by the theorem.
Dick said:The function is monotone increasing, it does have an inverse even if the derivative happens to be zero at point. And you don't have to evaluate f'(pi/2) to solve the problem, you have to find f'(f^(-1)(-1)). What's f^(-1)(-1)?
Shackleford said:I know I'm not evaluating at that point. To me, it seems that the theorem states that for every x in the interval the derivative of the function cannot be zero. Pi/2 is in that interval.
Shackleford said:The answer to the problem is (c).
Dick said:That's not what it says. It's says f' has to be continuously differentiable and nonzero NEAR the point f^(-1)(-1). Not everywhere.
Shackleford said:What are you talking about? It says it right here.
http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20111109_161232.jpg
I'm interpreting I = (0, 2pi).
Dick said:You don't need to take I=(0,2pi). f^(-1)(-1)=pi, so take I=(pi-1/4,pi+1/4). Or any other small interval around pi that doesn't include pi/2. Then it fits your statement, doesn't it? That's what I mean by NEAR pi.