Applications Of Derivatives Question 168
Question: Let f(x) be a twice differentiable function for all real values of x and satisfies f(1)=1, f(2)=4, f(3)=9. Then which of the following is definitely true-
Options:
A) $ f’’(x)=2\forall x\in (1,3) $
B) $ f’’(x)=f’(x)=5 $ for some $ x\in (2,3) $
C) $ f’’(x)=3,\forall ,x\in (2,3) $
D) $ f’’(x)=2 $ for some $ x\in (1,3) $
Show Answer
Answer:
Correct Answer: D
Solution:
[d] Let $ g(x)=f(x)-x^{2}. $ We have $ g(1)=0,g(2)=0,g(3)=0 $
$ [\therefore f(1)=1,f(2)=4,,f(3)=9] $ . From Rolle’s theorem on $ g(x),g’(x)=0 $ for at least $ x\in (1,2). $ Let $ g’(c_1)=0 $ where $ c_1\in (1,2) $ . Similarly, g(x) =0 for at least one $ x\in (2,3). $ Let $ g’(c_2)=0 $ Where $ c_2\in (2,3) $ . Therefore, $ g’(c_1)=g’(c_2)=0 $ By Rolle’s Theorem at least one $ x\in (c_1,c_2) $ such that $ g’’(x)=0 $ or $ f’’(x)=2 $ for some $ x\in (1,3) $
BETA
