The discussion revolves around a proof related to the Mean Value Theorem, specifically concerning a differentiable function defined on the interval [0,1] with given derivative conditions at the endpoints. The original poster presents a statement that claims the existence of a point c in (0,1) where the derivative equals the function value.
The conversation reflects differing interpretations of the statement's validity. Some participants assert that the existence of a counterexample indicates the statement is false, while others clarify the distinction between a statement being unprovable and being false. The dialogue suggests an ongoing examination of the assumptions involved.
There is mention of potential missing conditions, such as the values of the function at the endpoints, which may affect the validity of the original claim.
gruba said:Homework Statement
Let f is differentiable function on [0,1] and f^{'}(0)=1,f^{'}(1)=0. Prove that \exists c\in(0,1) : f^{'}(c)=f(c).
Homework Equations
-Mean Value Theorem
The Attempt at a Solution
The given statement is not true. Counter-example is f(x)=\frac{2}{\pi}\sin\frac{\pi}{2}x+10.
Does this mean that the statement can't be proved?