Let $f: mathbb{R} rightarrow mathbb{R}$ be a differentiable function such that $f(1)=f(0)+1$. Then which of the following is/are true?
(a) $f^{prime}$ is constant.
(b) $f(2)=f(1)+1$
(c) $f^{prime}(x)=1$ for some $x$ in $(0,1)$ .
(d) $left|f^{prime}(x)right| leq 1$ for all $x$ in $(0,1)$
Source: Indian competitive exam for M.Sc in Applied Mathematics.
We see that $f(x)=x+C$, where $C$ is a constant satisfies the hypothesis of the problem.Hence all the options will be true.
On the other hand $f(x)=sinleft(frac{pi x}{2}right)$ also satisfies the hypothesis of the problem. But this function satisfies only option $(c)$.
