Limits Continuity And Differentiability Question 26
Question: Let f be a function which is continuous and differentiable for all real x. If $ f(2)=-4 $ and $ f’(x)\ge 6 $ for all $ x\in [2,4], $ then
Options:
A) $ f(4)<8 $
B) $ f(4)\ge 8 $
C) $ f(4)\ge 12 $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
By mean value theorem, there exists a real number $ c\in (2,4) $ such that $ f’(c)=\frac{f(4)-f(2)}{4-2}\Rightarrow f’(c)=\frac{f(4)+4}{2} $ Since, $ f’(c)\ge 6,\forall x\in [2,4] $
$ \therefore f’(c)\ge 6,\Rightarrow \frac{f(4)+4}{2}\ge 6 $
$ \Rightarrow f(4)+4\ge 12\Rightarrow f(4)\ge 8 $ .
BETA
