# Let be the set of all functions

Question:

Let $\mathrm{S}$ be the set of all functions $f:[0,1] \rightarrow R$, which are continuous on $[0,1]$ and differentiable on $(0,1)$. Then for every $f$ in $S$, there exists a $c \in(0,1)$, depending on $f$, such that:

1. (1) $|f(c)-f(1)|<(1-c)\left|f^{\prime}(c)\right|$

2. (2) $\frac{f(1)-f(c)}{1-c}=f^{\prime}(c)$

3. (3) $|f(c)+f(1)|<(1+c)\left|f^{\prime}(c)\right|$

4. (4) None of these

Correct Option: , 4

Solution:

For a constant function $f(x)$, option (1), (3) doesn't hold and by LMVT theorem, option (2) is incorrect.