Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression $${{{x^m}{y^n}} \over {\left( {1 + {x^{2m}}} \right)\left( {1 + {y^{2n}}} \right)}}$$ is :
JEE · Math · previous-year question
- A.$${1 \over 2}$$
- B.$${1 \over 4}$$correct
- C.$${{m + n} \over {6mn}}$$
- D.1
Answer
B. $${1 \over 4}$$
Explanation
$${{{x^m}{y^n}} \over {\left( {1 + {x^{2m}}} \right)\left( {1 + {y^{2n}}} \right)}} = {1 \over {\left( {{x^m} + {1 \over {{x^m}}}} \right)\left( {{y^n} + {1 \over {{y^n}}}} \right)}} \le {1 \over 4}$$ using AM $$ \ge $$ GM
Practice more JEE questions
Answer thousands more real JEE previous-year questions free, get graded instantly, and climb the global ranked leaderboard.
Practice JEE free →More JEE Math questions
- What is the total number of distinct x \in \mathbb{R} for which \left|\begin{array}{ccc}x …
- Let m be the smallest positive integer such that the coefficient of x^{2} in the expansion…
- What is the total number of distinct x \in[0,1] for which \int_{0}^{x} \frac{t^{2}}{1+t^{4…
- Let \alpha, \beta \in \mathbb{R} be such that \lim _{x \rightarrow 0} \frac{x^{2} \sin (\b…
- Let z=\frac{-1+\sqrt{3} i}{2}, where i=\sqrt{-1}, and r, s \in\{1,2,3\}. Let P=\left[\begi…
- For how many values of p, the circle x^{2}+y^{2}+2 x+4 y-p=0 and the coordinate axes have …
- Let f: \mathbb{R} \rightarrow \mathbb{R} be a differentiable function such that f(0)=0, f\…
- For a real number \alpha, if the system \[ \left[\begin{array}{ccc} 1 & \alpha & \alpha^{2…