Limits Continuity And Differentiability Question 47
Question: Which one of the following statements is correct in respect of the function $ f(x)=x^{3}\sin x $ ?
Options:
A) f’(x) changes sign from positive to negative at x = 0
B) f ‘(x) changes sign from negative to positive at x = 0
C) Does not change sign at x = 0
D) $ f’’(0)\ne 0 $
Show Answer
Answer:
Correct Answer: C
Solution:
$ f(x)=x^{3}\sin x $
$ f’(x)=3x^{2}\sin x+x^{3}\cos x $
$ f’(x)=0 $
$ \Rightarrow 3x^{2}\sin x+x^{3}\cos x=0 $
$ \Rightarrow x^{2}(3\sin x+x\cos x)=0 $
$ \Rightarrow x=0,3\sin x+x\cos x=0….(1) $ Put $ x=0 $ in (1) $ 3\sin x=0\Rightarrow \sin x=0 $
$ {f^{\centerdot }}(x)=6x\sin x+3x^{2}\cos x+3x^{2}\cos x+x^{3}(-\sin x) $
$ f’’(0)=0 $
BETA
