A) \[\frac{529}{49}\]
B) \[\frac{8}{89}\]
C) \[\frac{49}{543}\]
D) None of these
Correct Answer: C
Solution :
Consider the function \[f(x)=\frac{{{x}^{2}}}{({{x}^{3}}+200)}\] .....(i) \[f'(x)=x\frac{(400-{{x}^{3}})}{{{({{x}^{3}}+200)}^{2}}}=0\] When \[x={{(400)}^{1/3}}\ ,\ (\because x\ne 0)\] \[x={{(400)}^{1/3}}-h\Rightarrow f'(x)>0\] \[x={{(400)}^{1/3}}+h\Rightarrow f'(x)<0\] \[\therefore \]\[f(x)\] has maxima at \[x={{(400)}^{1/3}}\] Since \[7<{{(400)}^{1/3}}<8,\]either \[{{a}_{7}}\]or \[{{a}_{8}}\]is the greatest term of the sequence. \[\because {{a}_{7}}=\frac{49}{543}\]and\[{{a}_{8}}=\frac{8}{89}\]and \[\frac{49}{543}>\frac{8}{89}\] \[\therefore \] \[{{a}_{7}}=\frac{49}{543}\]is the greatest term.
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