Paragraph: Read the following passage and answer the questions. For every function $f(x)$ which is twice differentiable, these will be good approximation of $\int_a^b f(x) d x \cong\left(\frac{b-a}{2}\right)\{f(a)+f(b)\}$. Now, if we take $c=\frac{a+b}{2}$, then using above again, we get $\int_a^b f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x \cong \frac{b-a}{4}\{f(a)+f(b)+2 f(c)\}$ and so on. We get approximation for value of $\int_a^b f(x) d x$.Question: If $f^{\prime \prime}(x) < 0, \forall x \in(a, b), c(c, f(c))$ is point of maxima where $c \in(a, b)$, then $f^{\prime}(c)$ is

• A

  $\frac{f(b)-f(a)}{b-a}$
• B

  $3\left(\frac{f(b)-f(a)}{b-a}\right)$
• C

  $2\left(\frac{f(b)-f(a)}{b-a}\right)$
• Correct answer
  D

  0

Hence, the correct answer is option: 'D'.