Question
Which one of the following statements is correct in respect of the function $$f\left( x \right) = {x^3}\sin \,x\,?$$
A.
$$f'\left( x \right)$$ changes sign from positive to negative at $$x = 0$$
B.
$$f'\left( x \right)$$ changes sign from positive to negative to positive at $$x = 0$$
C.
does not change sign at $$x = 0$$
D.
$$f''\left( 0 \right) \ne 0$$
Answer :
does not change sign at $$x = 0$$
Solution :
$$\eqalign{
& f\left( x \right) = {x^3}\sin \,x \cr
& f'\left( x \right) = 3{x^2}\sin \,x + {x^3}\cos \,x \cr
& f'\left( x \right) = 0 \cr
& \Rightarrow 3{x^2}\sin \,x + {x^3}\cos \,x = 0 \cr
& \Rightarrow {x^2}\left( {3\,\sin \,x + x\,\cos \,x} \right) = 0 \cr
& \Rightarrow x = 0,\,3\,\sin \,x + x\,\cos \,x = 0.....(1) \cr
& {\text{Put }}x = 0,{\text{ in equation }}(1) \cr
& 3\,\sin \,x = 0\,\,\, \Rightarrow \sin \,x = 0 \cr
& f''\left( x \right) = 6x\,\sin \,x + 3{x^2}\cos \,x + 3{x^2}\cos \,x + {x^3}\left( { - \sin \,x} \right) \cr
& f''\left( 0 \right) = 0 \cr} $$