QJEEM20Ma9b34: JEE Main 2020 Mathematics - Class 12 JEEM20Ma9b34 Solution: Let S be the set of all functions ƒ : [0,1] R, which are…

Question

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Let S be the set of all functions ƒ : [0,1]

\to

R, which are continuous on [0,1] and differentiable on (0,1). Then for every ƒ in S, there exists a c

\in

(0,1), depending on ƒ, such that

Accepted Answer

If we consider the case where f(x) is a constant function, then its derivative f'(x) is equal to 0 for all x in the interval (0,1).

Therefore, if we substitute this into the expressions provided in Options A, B and C, we would have :

Option A : |f(c) - f(1)| < |f'(c)| would become |constant - constant| < |0|, which is 0 < 0. This is not true.

Option B : |f(c) + f(1)| < (1 + c)|f'(c)| would become |constant + constant| < (1 + c) $\times$ 0, which is a positive number < 0. This is not true.

Option C : |f(c) - f(1)| < (1 - c)|f'(c)| would become |constant - constant| < (1 - c) $\times$ 0, which is 0 < 0. This is not true.

Hence, for the case where f(x) is a constant function, none of the options A, B and C are correct.

So, the correct answer would be Option D : None.

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