question_answer

The number of solution of the equation \[{{x}^{3}}+{{x}^{2}}+4x+2\,\sin \,x=0\] is

A)  2                                

B)  3                

C)  1                                

D)  8

Correct Answer: C

Solution :

 \[f(x)={{x}^{3}}+{{x}^{2}}+4x+2\,\,\sin \,\,x=0\] \[f'(x)=3{{x}^{2}}+2x+4+2\,\,\cos \,\,x\] For \[\cos \,\,x=\pm \,1\] \[f'(x)=3{{x}^{2}}+2x+4\pm 2\] \[\Rightarrow \]            \[(\because \,\,D={{b}^{2}}-4ac)\] Hence, only one solution exist, i.e., \[x=0\].