JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(x, y\) be positive real numbers and \(m, n\) positive integers. The maximum value of the expression \(\frac{{{x^m}{y^n}}}{{\left( {1 + {x^{2m}}} \right)\,\left( {1 + {y^{2n}}} \right)}}\) is
- A \(1\)
- B \(\frac{1}{2}\)
- C \(\frac{1}{4}\)
- D \(\frac{{m + n}}{{6mn}}\)
Answer & Solution
Correct Answer
(C) \(\frac{1}{4}\)
Step-by-step Solution
Detailed explanation
\(\frac{{{x^m}{y^n}}}{{\left( {1 + {x^{2m}}} \right)\left( {1 + {y^{2n}}} \right)}}\) \( = \frac{1}{{\left( {{x^m} + \frac{1}{{{x^m}}}} \right)\left( {{y^n} + \frac{1}{{{y^n}}}} \right)}}{\rm{ }}\) \({\rm{Put\,\, }}{x^m} + \frac{1}{{{x^m}}} \ge 2\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The square of the distance of the point \((-2, -8, 6)\) from the line \(\dfrac{x-1}{1} = \dfrac{y-1}{2} = \dfrac{z}{-1}\) along the line \(\dfrac{x+5}{1} = \dfrac{y+5}{-1} = \dfrac{z}{2}\) is equal to:JEE Mains 2026 Hard
- Let \(A=\left(\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right),\) where \([t]\) denotes the greatest integer less than or equal to \(\mathrm{t}\). If \(\operatorname{det}(\mathrm{A})=192\), then the set of values of \(\mathrm{x}\) is the intervalJEE Mains 2021 Hard
- The number of real solutions of the equation \(3\left(x^2+\frac{1}{x^2}\right)-2\left(x+\frac{1}{x}\right)+5=0\), isJEE Mains 2023 Hard
- Let \(A=\left(\begin{array}{cc}1+ i & 1 \\ - i & 0\end{array}\right)\) where \(i =\sqrt{-1}\) Then, the number of elements in the set \(\left\{ n \in\{1,2, \ldots, 100\}: A ^{ n }= A \right\}\) isJEE Mains 2022 Hard
- \(\int_{0}^{2}\left(\left|2 x^{2}-3 x\right|+\left[x-\frac{1}{2}\right]\right) d x\),where is the greatest integer function, is equal to.JEE Mains 2022 Hard
- Tangents are drawn to the hyperbola \(4{x^2} - {y^2} = 36\) at the points \(P\) and \(Q.\) If these tangents intersect at the point \(T(0,3)\) then the area (in sq. units) of \(\Delta PTQ\) is :JEE Mains 2018 Hard
More PYQs from JEE Mains
- If an equation of a tangent to the curve, \(y = \cos \,\left( {x + f} \right),\, - 1\, - \pi \le x \le 1 + \pi ,\) is \(x + 2y = k\) then \(k\) is equal toJEE Mains 2013 Hard
- Let \(A\) be a \(3 \times 3\) matrix such that \(\operatorname{adj} A=\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1\end{array}\right]\) and \(B = adj (\) adj \(A )\) If \(| A |=\lambda\) and \(\left|\left( B ^{-1}\right)^{ T }\right|=\mu,\) then the ordered pair \((|\lambda|, \mu)\) is equal toJEE Mains 2020 Hard
- Let \(f, g:(0, \infty) \rightarrow R\) be two functions defined by \(f(x)=\int_{-x}^x\left(|t|-t^2\right) e^{-t^2} d t\) and \(g(x)=\int_0^{x^2} t^{1 / 2} e^{-t} d t\). Then the value of \(\left(\mathrm{f}\left(\sqrt{\log _{\mathrm{e}} 9}\right)+\mathrm{g}\left(\sqrt{\log _{\mathrm{e}} 9}\right)\right)\) isJEE Mains 2024 Hard
- The area (in sq. units) of the region \(A=\{(x, y):(x-1)[x] \leq y \leq 2 \sqrt{x}, 0 \leq x \leq 2\}\) where \([t]\) denotes the greatest integer function, isJEE Mains 2020 Hard
- Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function defined \(f(x)=\frac{x}{\left(1+x^4\right)^{1 / 4}}\) and \(g(x)=f(f(f(f(x))))\) then \(18 \int_0^{\sqrt{2 \sqrt{5}}} x^2 g(x) d x\)JEE Mains 2024 Hard
- For \(x\, \in \,R\,,\,x\, \ne \, - 1,\) if \({(1 + x)^{2016}} + x{(1 + x)^{2015}} + {x^2}{(1 + x)^{2014}} + ....{x^{2016}} = \sum\limits_{i = 0}^{2016} {{a_i\,}{\,x^i}} ,\) then \(a_{17}\) is equal toJEE Mains 2016 Hard