A) 0
B) ? 1
C) \[6\,\,\log {{}_{e}}\,2\]
D) 6
Correct Answer: A
Solution :
\[y={{x}^{3}}\log {{\log }_{e}}(1+x)\] Þ \[{y}'=3{{x}^{2}}\log {{\log }_{e}}\,(1+x)+\frac{{{x}^{3}}}{1+x}.\frac{1}{{{\log }_{e}}(1+x)}\] Þ \[{y}''=6x\log {{\log }_{e}}(1+x)+\frac{3{{x}^{2}}}{{{\log }_{e}}(1+x)}.\frac{1}{(1+x)}\] \[-\frac{{{x}^{3}}}{{{(1+x)}^{2}}{{\log }_{e}}(1+x)}-\frac{{{x}^{3}}}{{{(1+x)}^{2}}}.\frac{1}{{{[{{\log }_{e}}(1+x)]}^{2}}}+\frac{3{{x}^{2}}}{(1+x){{\log }_{e}}(1+x)}\] Þ \[{y}''(0)=0\].
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