question_answer

Let  f : R\[\to \]R be given by:

\[f\,(x)\left\{ \begin{matrix}    {{x}^{5}}+5{{x}^{4}}+10{{x}^{3}}+3x+1 & x\,<\,0  \\    {{x}^{2}}-x+1, & 0\,\,\le \,x\,<\,1;  \\    \frac{2}{3}{{x}^{3}}-4{{x}^{2}}+7x-\frac{8}{3}, & 1\,\le \,x\,<\,3  \\    (x\,-\,2)\,{{\log }_{e\,}}\,(x-2)\,-\,x+\frac{10}{3}, & x\,\,\ge \,3  \\ \end{matrix} \right.\]

Then which of the following options is/are correct?

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

B) \[f\] is onto

C) \[f\] has a local maximum at \[x=1\]

D) \[f\] is not differentiable at \[x=1\]

Correct Answer: B , C , D

Solution :

\[f(x)=\left[ \begin{matrix}    {{x}^{5}}+5{{x}^{4}}+10{{x}^{3}}+10{{x}^{2}}+3x+1 & x

\[\therefore \]range = R \[(In(x-2)\]contains all real values)

\[f'(x)=\left[ \begin{matrix}    5{{x}^{4}}+20{{x}^{3}}+30{{x}^{2}}+20x+3 & x

\[f''(x)=\left[ \begin{matrix}    20{{x}^{3}}+60{{x}^{2}}+60x+20 & x

\[f''(x)\left[ \begin{matrix}    20{{(1+x)}^{3}} & x

Solution :

Same as above

Solution :

Same as above