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Let \( \mathrm{S} \) be the set of all function \( f:[0,1] \rightarrow \mathrm{R} \), which are continuous on \( [0,1] \…

Q1

Let \( \mathrm{S} \) be the set of all function \( f:[0,1] \rightarrow \mathrm{R} \), 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

[JEE Main 2020, 8 Jan (Shift 2)]

a

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

b

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

c

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

d

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

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