IIT-JEE 1989Question 1
State TRUE or FALSE: If $x$ and $y$ are positive real numbers and $m,n$ are any positive integers, then $\dfrac{x^{n}y^{m}}{(1+x^{2n})(1+y^{2m})}>\dfrac14$.
Sequence and Series · Means and Inequalities (AM, GM, HM)
Solution
Formulas used. (1) AM-GM: $1+t^2\ge2t$ for $t>0$, equality at $t=1$Step 1 - Each factor. With $t=x^n$: $\dfrac{x^n}{1+x^{2n}}\le\dfrac12$. Likewise $\dfrac{y^m}{1+y^{2m}}\le\dfrac12$. Step 2 - Product. $$\frac{x^ny^m}{(1+x^{2n})(1+y^{2m})}\le\frac14 .$$ Step 3. The expression is never greater than $\frac14$ (for $x=y=1$ it equals $\frac14$). So the statement is FALSE. Answer: (B)