Let S be the set of all functions

Question:

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

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

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

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

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

Correct Option: , 2

Solution:

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