Sequences And Series Ques 20
- If $x$ and $y$ are positive real numbers and $m, n$ are any positive integers, then $\frac{x^n y^m}{\left(1+x^{2 n}\right)\left(1+y^{2 m}\right)}>\frac{1}{4}$.
Show Answer
Answer:
Correct Answer: 20.(False)
Solution: Using $\mathrm{AM} \geq \mathrm{GM}$,
$\quad \frac{1+x^{2 n}}{2} \geq \sqrt{1 \cdot x^{2 n}} $
$\Rightarrow \quad \frac{1+x^{2 n}}{2} \geq x^n $
$\Rightarrow \quad \frac{x^n}{1+x^{2 n}} \leq \frac{1}{2} $
$\therefore \quad \frac{x^n \cdot y^n}{\left(1+x^{2 n}\right)\left(1+y^{2 m}\right)} \leq \frac{1}{4}$
Hence, it is false statement.
BETA
