question_answer

If \[f(x)=x\left| \,\,x\,\, \right|-\ell n\,\,{{x}^{2}},\] then which statement is not true?

A) \[f(x)\] is increasing in \[(-\infty ,0)\]

B) \[f(x)\] is decreasing in \[(0,\infty )\]

C) \[f(x)\] is increasing m \[(1,\infty )\]

D) \[f(x)\] is decreasing in (0, 1)

Correct Answer: B

Solution :

\[f(x)=\left[ \begin{matrix}    {{x}^{2}}-\ell n{{x}^{2}}, & x\ge 0  \\    -{{x}^{2}}-\ell n{{x}^{2}}, & x<0  \\ \end{matrix} \right.\]

\[f'(x)=\left[ \begin{matrix}    2x-2/x, & x>0  \\    -\,2x-2/x, & x<0  \\ \end{matrix} \right.\]

\[=\left[ \begin{matrix}    \frac{2\,\,(x-1)\,\,(x+1)}{x}, & x>0  \\    \frac{-\,2({{x}^{2}}+1)}{x}, & x<0  \\ \end{matrix} \right.\]