Limits Continuity And Differentiability Question 118
Question: If $ f’’(x)<0,\forall x\in (a,b), $ then $ f’(x)=0 $ occurs
Options:
A) Exactly once in (a, b)
B) At most once in (a, b)
C) At least once in (a, b)
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
Suppose, there are two points $ x_1 $ and $ x_2 $ in (a, b) such that $ f’(x_1)=f’(x_2)=0 $ .
By Rolle’s theorem applied to f? on $ [x_1,x_2] $ , there must be a $ c\in (x_1,x_2) $ such that $ f’’(c)=0 $ .
This contradicts the given condition $ f’’(x)<0,\forall x\in (a,b) $ .
Hence, our assumption is wrong. Therefore, there can be at most one point in (a, b) at which f?(x) is zero.
BETA
