Let S be the set of all functions f:[0,1]R, which are continuous on [0,1] and differentiab…

Question

Let $S$ be the set of all functions $f:[0,1]	o\mathbb{R}$, 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:

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) ( 	imes \]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) ( 	imes \]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.

Solutions · 1