JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let the set of all positive values of \(\lambda\), for which the point of local minimum of the function \(\left(1+x\left(\lambda^2-x^2\right)\right)\) satisfies \(\frac{x^2+x+2}{x^2+5 x+6}<0\), be \((\alpha, \beta)\). Then \(\alpha^2+\beta^2\) is equal to ...........
- A \(13\)
- B \(40\)
- C \(39\)
- D \(50\)
Answer & Solution
Correct Answer
(C) \(39\)
Step-by-step Solution
Detailed explanation
\( \frac{x^2+x+2}{x^2+5 x+6}<0 \) \( \Rightarrow \frac{1}{(x+2)(x+3)}<0\) \( \mathrm{x} \in(-3,-2) \ldots \ldots \ldots . .(1) \) \( \mathrm{f}(\mathrm{x})=1+\mathrm{x}\left(\lambda^2-\mathrm{x}^2\right)\) Finding local minima…
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 value of the integral \(\int \limits_1^2\left(\frac{t^4+1}{t^6+1}\right) d t\) is \(..........\).JEE Mains 2023 Hard
- Let \(\lambda, \mu \in R\). If the system of equations \( 3 x+5 y+\lambda z=3 \) \( 7 x+11 y-9 z=2 \) \( 97 x+155 y-189 z=\mu\) has infinitely many solutions, then \(\mu+2 \lambda\) is equal to :JEE Mains 2024 Hard
- The points \(\left( {0,\frac{8}{3}} \right),\,(1,3)\) and \((82,30)\)JEE Mains 2015 Hard
- Let \(\mathrm{P}(4,4 \sqrt{3})\) be a point on the parabola \(y^2=4 \mathrm{a} x\) and PQ be a focal chord of the parabola. If M and \(N\) are the foot of perpendiculars drawn from \(P\) and \(Q\) respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :JEE Mains 2025 Medium
- If \(x \phi(x)=\int_{5}^{x}\left(3 t^{2}-2 \phi^{\prime}(t)\right) d t, x\,>\,-2\), and \(\phi(0)=4\) then \(\phi(2)\) is .... .JEE Mains 2021 Hard
- The number of points of intersection of \(| z -(4+3 i )|=2\) and \(| z |+| z -4|=6, z \in C\) isJEE Mains 2022 Medium
More PYQs from JEE Mains
- Let \(A =\{0,1,2, \ldots 9)\). Let R be a relation on A defined by \(( x , y ) \in R\) if and only if \(| x - y |\) is a multiple of 3 .
Given below are two statements:
Statement I: \(n ( R )=36\)
Statement II: \(R\) is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given belowJEE Mains 2026 Hard - Let the area of the triangle with vertices \(A (1, \alpha)\), \(B (\alpha, 0)\) and \(C (0, \alpha)\) be \(4\, sq.\) units. If the point \((\alpha,-\alpha),(-\alpha, \alpha)\) and \(\left(\alpha^{2}, \beta\right)\) are collinear, then \(\beta\) is equal toJEE Mains 2022 Medium
- Let \(S\) be the set of all the natural numbers, for which the line \(\frac{x}{a}+\frac{y}{b}=2\) is a tangent to the curve \(\left(\frac{ x }{ a }\right)^{ n }+\left(\frac{ y }{ b }\right)^{ n }=2\) at the point \(( a , b ), ab \neq 0\). ThenJEE Mains 2022 Medium
- Let \(\overrightarrow{ a }=\alpha \hat{ i }+2 \hat{ j }-\hat{ k }\) and \(\overrightarrow{ b }=-2 \hat{ i }+\alpha \hat{ j }+\hat{ k }\), where \(\alpha \in R\). If the area of the parallelogram whose adjacent sides are represented by the vectors \(\vec{a}\) and \(\vec{b}\) is \(\sqrt{15\left(\alpha^{2}+4\right)}\), then the value of \(2|\vec{a}|^{2}+(\vec{a} \cdot \vec{b})|\vec{b}|^{2}\) is equal toJEE Mains 2022 Hard
- If the function \(f(x)=\frac{1}{x} \log _{e}(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}) , \quad x<0\) \(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad k \quad, \quad x=0\) \(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1} ,\,\,\, x>0\) is continuous at \(x=0\), then \(\frac{1}{a}+\frac{1}{b}+\frac{4}{k}\) is equal to :JEE Mains 2021 Hard
- If the solution curve \(y=y(x)\) of the differential equation \(\left(1+y^2\right)\left(1+\log _6 x\right) d x+x d y=0, x>0\) passes through the point \((1,1)\) and \(y(\mathrm{e})=\frac{\alpha-\tan \left(\frac{3}{2}\right)}{\beta+\tan \left(\frac{3}{2}\right)}\), then \(\alpha+2 \beta\) is ...........JEE Mains 2024 Medium