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

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 $n=7$.