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) \[f''(0)\ne 0\]
Correct Answer: C
Solution :
[c] \[f(x)={{x}^{3}}\sin x\] \[f'(x)=3{{x}^{2}}\sin x+{{x}^{3}}\cos x\] \[f'(x)=0\] \[\Rightarrow 3{{x}^{2}}\sin x+{{x}^{3}}\cos x=0\] \[\Rightarrow {{x}^{2}}(3\,\,\sin x+x\cos x)=0\] \[\Rightarrow x=0,\,\,3\sin x+x\cos x=0....(1)\] Put \[x=0\] in (1) \[3\sin x=0\Rightarrow \sin x=0\] \[{{f}^{\centerdot }}(x)=6x\sin x+3{{x}^{2}}\cos x+3{{x}^{2}}\cos x+{{x}^{3}}(-\sin x)\]\[f''(0)=0\]
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