Question

Statement-I : The largest term in the sequence $a_n=\frac{n^2}{n^3+200}, n \in N$ is the $7^{\text {th }}$ term. Because
Statement-II : The function $f ( x )=\frac{ x ^2}{ x ^3+200}$ attains local maxima at $x =7$.

• A. Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I.
• B. Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-I.
• C. Statement-I is true, Statement-II is false.
• D. Statement-I is false, Statement-II is true.

Answer 1

Answer: (C)

St. II :- $f(x)=\frac{x^2}{x^3+200}$
$f^{\prime}(x)=\frac{2 x\left(x^3+200\right)-3 x^4}{\left(x^3+200\right)^2}=\frac{x\left(400-x^3\right)}{\left(x^3+200\right)^2}$

St. II is false.
St. I $\because f ( x )$ has maxima at $x =(400)^{1 / 3}$ & 7 is the closest natural number.
$\therefore a _{ n }$ has greatest value for $n =7$.