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 :
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