JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be a twice differentiable function such that \((\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x\) \(\sin \mathrm{y})(f(2 \mathrm{x}+2 \mathrm{y})+f(2 \mathrm{x}-2 \mathrm{y}))\), for all \(\mathrm{x}, \mathrm{y} \in \mathbf{R}\).
If \(f^{\prime}(0)=\frac{1}{2}\), then the value of \(24 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)\) is:
- A 2
- B \(-3\)
- C 3
- D \(-2\)
Answer & Solution
Correct Answer
(B) \(-3\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & (\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x \sin y) \\ & (f(2 x+2 y)+f(2 x-2 y)) \\ & f(2 x+2 y)(\sin (x-y))=f(2 x-2 y) \sin (x+y) \\ & \frac{f(2 x+2 y)}{\sin (x+y)}=\frac{f(2 x-2 y)}{\sin (x-y)} \\ & \text { Put } 2 x+2 y=m, 2 x-2 y=n \\ & \frac{f(m)}{\sin…
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
- If \(\frac{ dy }{ dx }+2 y \tan x =\sin x , 0< x <\frac{\pi}{2}\) and \(y \left(\frac{\pi}{3}\right)=\) 0 , then the maximum value of \(y(x)\) is.JEE Mains 2022 Hard
- The lines \(x=a y-1=z-2\) and \(x=3 y-2=b z-2,(a b \neq 0)\) are coplanar, if:JEE Mains 2021 Medium
- The number of real roots of the equation \(\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{4}\) is:JEE Mains 2021 Hard
- Let \({\left( { - \,2\, - \,\frac{1}{3}\,i} \right)^3} = \frac{{x \,+ \,iy}}{{27}}(i\, = \,\sqrt { - 1} ),\) where \(x\) and \(y\) are real numbers, then \(y -x\) equalsJEE Mains 2019 Hard
- If \(5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9,\) then \(\cos 4x\) is equal toJEE Mains 2017 Hard
- If \(A> 0, B > 0\) and \(A + B = \frac{\pi }{6}\), then the minimum value of \(tan\,A + tan\,B\) isJEE Mains 2016 Hard
More PYQs from JEE Mains
- If \(0 \le x \le \pi \) and \({81^{{{\sin }^2}x}} + {81^{{{\cos }^2}x}} = 30\), then \(x =\)JEE Mains 2021 Hard
- If the shortest distance between the straight lines \(3(x-1)=6(y-2)=2(z-1)\) and \(4(\mathrm{x}-2)=2(\mathrm{y}-\lambda)=(\mathrm{z}-3), \lambda \in \mathrm{R}\) is \(\frac{1}{\sqrt{38}}\), then the integral value of \(\lambda\) is equal to :JEE Mains 2021 Medium
- If a directrix of a hyperbola centered at the origin and passing through the point \((4, -2\sqrt 3)\) is \(5x = 4\sqrt 5\) and its eccentricity is \(e\), thenJEE Mains 2019 Hard
- From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :JEE Mains 2025 Hard
- Consider the sets \(\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+\mathrm{y}^2=25\right\}\), \(\mathrm{B}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+9 \mathrm{y}^2=144\right\}, \mathrm{C}=\{(\mathrm{x}, \mathrm{y})\) \(\left.\in \mathbb{Z} \times \mathbb{Z}: x^2+y^2 \leq 4\right\}\), and \(D=A \cap B\). The total number of one-one functions from the set D to the set C is:JEE Mains 2025 Hard
- Let \(z_{1}, z_{2}\) be the roots of the equation \(z^{2}+a z+\) \(12=0\) and \(z _{1}, z _{2}\) form an equilateral triangle with origin. Then, the value of \(| a |\) isJEE Mains 2021 Hard