JEE Mains · Maths · STD 12 - 6. Application of derivatives
If the local maximum value of the function \(f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^2 x}, \quad x \in\left(0, \frac{\pi}{2}\right)\), is \(\frac{k}{e}\), then \(\left(\frac{ k }{ e }\right)^8+\frac{ k ^8}{ e ^5}+ k ^8\) is equal to
- A \(e^5+e^6+e^{11}\)
- B \(e^3+e^5+e^{11}\)
- C \(e ^3+ e ^6+ e ^{11}\)
- D \(e^3+e^6+e^{10}\)
Answer & Solution
Correct Answer
(C) \(e ^3+ e ^6+ e ^{11}\)
Step-by-step Solution
Detailed explanation
\(\text { Let } y=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^2 x}\) \(\ln y=\sin ^2 x \cdot \ln \left(\frac{\sqrt{3 e}}{2 \sin x}\right)\)…
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
- Four fair dice are thrown independently \(27\) times. Then the expected number of times, at least two dice show up a three or a five, isJEE Mains 2020 Hard
- The shortest distance between the lines \(\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}\) and \(x+y+z+1=0\), \(2 x-y+z+3=0\) isJEE Mains 2020 Hard
- The number of points, where the curve \(y=x^5-20 x^3+50 x+2\) crosses the \(x\)-axis, is \(............\).JEE Mains 2023 Hard
- Let \(a, b, c > 1, a^3, b^3\) and \(c^3\) be in \(A.P.\), and \(\log _a b\), \(\log _c a\) and \(\log _b c\) be in G.P. If the sum of first \(20\) terms of an \(A.P.\), whose first term is \(\frac{a+4 b+c}{3}\) and the common difference is \(\frac{a-8 b+c}{10}\) is \(-444\), then abc is equal toJEE Mains 2023 Hard
- If \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly \(7\) solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for the least value of \(n \in N\) then \(\sum_{\mathrm{k}=1}^{\mathrm{n}} \frac{\mathrm{k}}{2^{\mathrm{k}}}\) is equal to :JEE Mains 2024 Hard
- Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If \(99\) more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly \(2\) balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle isJEE Mains 2019 Hard
More PYQs from JEE Mains
- Let \(z\) and \(w\) be two complex numbers such that \(w=z \bar{z}-2 z+2,\left|\frac{z+i}{z-3 i}\right|=1\) and \(\operatorname{Re}(w)\) has minimum value. Then, the minimum value of \(n \in N\) for which \(w ^{ n }\) is real, is equal to..........JEE Mains 2021 Hard
- The shortest distance between the \(z-\) axis and the line \(x + y + 2z - 3\, = 0 \,= 2x + 3y + 4z - 4\), isJEE Mains 2015 Hard
- Let \(\vec{a}=a_i \hat{i}+a_2 \hat{j}+a_3 \hat{k}\) and \(\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\) be two vectors such that \(|\vec{a}|=1 ; \quad \vec{a} \cdot \vec{b}=2\) and \(|\vec{b}|=4\). If \(\vec{c}=2(\vec{a} \times \vec{b})-3 \vec{b}\), then the angle between \(\vec{b}\) and \(\vec{c}\) is equal to :JEE Mains 2024 Hard
- If sum of the first \(21\) terms of the series \(\log _{9^{1 / 2}} x +\log _{9^{1 / 3}} x +\log _{9^{1 / 4}} x +\ldots ., x >0\) , where \(x>0\) is \(504,\) then \(\mathrm{x}\) is equal to:JEE Mains 2021 Medium
- Let \((a, b)\) be the point of intersection of the curve \(x^2=2 y\) and the straight line \(y-2 x-6=0\) in the second quadrant. Then the integral \(I=\int_a^b \frac{9 x^2}{1+5^x} d x\) is equal to :JEE Mains 2025 Medium
- Let an ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), \(a < b\), pass through the point \((4, 3)\) and have eccentricity \(\dfrac{\sqrt{5}}{3}\). Then the length of its latus rectum is :JEE Mains 2026 Medium