The problem involves proving the existence of a point using Rolle's Theorem, specifically in the context of a function \( f \) that is continuous on the interval \([a,b]\) and differentiable on \((a,b)\). The original poster presents a condition involving the squares of the function values at the endpoints of the interval.
The discussion is focused on clarifying the notation and ensuring that the correct form of the equation is being used. Participants are exploring the implications of the defined function \( g(x) \) and its suitability for applying Rolle's Theorem. There is a recognition of the need to establish the correct conditions for the theorem to be applicable.
There is some confusion regarding the notation used in the problem statement, particularly whether it involves a product or a difference of squares. This has led to a deeper examination of the assumptions underlying the application of Rolle's Theorem.
Dick said:I think you mean f(b)^2-f(a)^2=b^2-a^2. Define g(x)=f(x)^2-x^2. Can you apply Rolle's theorem to g(x) on the interval [a,b]?
Dick said:No. '^2' just means squared. E.g. f(b)^2=f(b)*f(b). Which is the same as [itex]f^2(b)[/itex]. The problem I'm having is that what you wrote looks like f(b)^2*f(a)^2=b^2-a^2. I really think it should be f(b)^2-f(a)^2. With a minus sign between the two squares, not a product of them.