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

<p>If we consider the case where f(x) is a constant function, then its derivative f&#39;(x) is equal to 0 for all x in the interval (0,1). </p> <p>Therefore, if we substitute this into the expressions provided in Options A, B and C, we would have :</p> <p>Option A : |f(c) - f(1)| &lt; |f&#39;(c)| would become |constant - constant| &lt; |0|, which is 0 &lt; 0. This is not true.</p> <p>Option B : |f(c) + f(1)| &lt; (1 + c)|f&#39;(c)| would become |constant + constant| &lt; (1 + c)$\times$0, which is a positive number &lt; 0. This is not true.</p> <p>Option C : |f(c) - f(1)| &lt; (1 - c)|f&#39;(c)| would become |constant - constant| &lt; (1 - c)$\times$0, which is 0 &lt; 0. This is not true.</p> <p>Hence, for the case where f(x) is a constant function, none of the options A, B and C are correct.</p> <p>So, the correct answer would be Option D : None.</p>