Question

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

• A. $\frac{f \left(1\right)-f \left(c\right)}{1-c}=f '\left(c\right)$
• B. $\left|f \left(c\right)-f \left(1\right)\right| < \left|f '\left(c\right)\right|$
• C. $\left|f \left(c\right)+f \left(1\right)\right| <\left(1+c\right) \left|f '\left(c\right)\right|$
• D. $\left|f \left(c\right)-f \left(1\right)\right| <\left(1-c\right) \left|f '\left(c\right)\right|$

Answer 1

Answer: (D)

Correct answer is (d) $\left|f \left(c\right)-f \left(1\right)\right| <\left(1-c\right) \left|f '\left(c\right)\right|$