Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20252 AprilMorning Shift
Exam
JEE Main
Year
2025
Date
2 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Differentiation

JEE Main 2025 · 2 April · Morning Shift · Mathematics · Differentiation

Let f: R arrow R be a twice differentiable function such that ( x...

Let f:R→Rbeatwicedifferentiablefunctionsuchthatf: \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})),forall, for all\mathrm{x}, \mathrm{y} \in \mathbf{R}.<br/>If.<br/>Iff^{\prime}(0)=\frac{1}{2},thenthevalueof, then the value of24 f^{\prime \prime}(5π3)\left(\frac{5 \pi}{3}\right) is:

Options

  1. A2
  2. B−3-3
  3. C3
  4. D−2-2

Correct answer

B. −3-3

Step-by-step solution

(sin⁡xcos⁡y)(f(2x+2y)−f(2x−2y))=(cos⁡xsin⁡y)(f(2x+2y)+f(2x−2y))f(2x+2y)(sin⁡(x−y))=f(2x−2y)sin⁡(x+y)f(2x+2y)sin⁡(x+y)=f(2x−2y)sin⁡(x−y) Put 2x+2y=m,2x−2y=nf(m)sin⁡(m2)=f(n)sin⁡(n2)=K⇒f(m)=Ksin⁡(m2)∴f(x)=Ksin⁡(x2)f′(x)=K2cos⁡(x2)PPut x=0;12=K2⇒K=1f′(x)=12cos⁡x2\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 \left(\frac{m}{2}\right)}=\frac{f(n)}{\sin \left(\frac{n}{2}\right)}=\mathrm{K} \\ & \Rightarrow f(m)=K \sin \left(\frac{m}{2}\right) \\ & \therefore f(x)=K \sin \left(\frac{x}{2}\right) \\ & f^{\prime}(x)=\frac{K}{2} \cos \left(\frac{x}{2}\right) \\ & P_{\text{Put }} x=0 ; \frac{1}{2}=\frac{K}{2} \Rightarrow K=1 \\ & f^{\prime}(x)=\frac{1}{2} \cos \frac{x}{2}\end{aligned}f′′(x)=−14sin⁡x24f′′(5π3)=(−14sin⁡(5π6))24=−248=−3\begin{aligned} & f^{\prime \prime}(x)=-\frac{1}{4} \sin \frac{x}{2} \\ & 4 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)=\left(-\frac{1}{4} \sin \left(\frac{5 \pi}{6}\right)\right) 24 \\ & =\frac{-24}{8}=-3\end{aligned}$
Practice Differentiation on Quantrex Academy →

More from Differentiation

Let ℝ denote the set of all real numbers. Consider the polynomial function f: ℝ arrow ℝ… JEE Advanced 2026Let f(x) and g(x) be twice differentiable functions satisfying f''(x) = g''(x) for all x… JEE Main 2026Let f be a real polynomial of degree n such that f(x) = f'(x) f''(x) , for all x ∈ ℝ . If… JEE Main 2026Let f(x)=x^ 3 +x^ 2 f^ (1)+2 x f^ (2)+f^ (3), x ∈ R . Then the value of f^ (5) is: JEE Main 2026Let ℝ denote the set of all real numbers. Let f: ℝ arrow ℝ and g: ℝ arrow(0,4) be… JEE Advanced 2025If _e y=3 ^ -1 x , then (1-x^2) y^ -x y^ at x=(1)/(2) is equal to JEE Main 2024Let f(x)=a x^3+b x^2+c x+41 be such that f(1)=40, f^ (1)=2 and f^ (1)=4 . Then… JEE Main 2024Suppose for a differentiable function h, h(0)=0, h(1)=1 and h^ (0)=h^ (1)=2 . If… JEE Main 2024If f(x)=\ array l x^3 ((1)/(x)), x ≠ 0 0 , x=0 array . then JEE Main 2024If y(θ)=(2 θ+ 2 θ)/( 3 θ+4 2 θ+5 θ+2) , then at θ=(π)/(2), y^ +y^ +y is equal to : JEE Main 2024Let f(x)=x^5+2 e^ x / 4 for all x ∈ R . Consider a function g(x) such that (g ∘ f)(x)=x… JEE Main 2024If y = ( x + 1)( x ^ 2 - x ) x x + x + x + (1)/(15)(3 ^ 2 x - 5) ^ 3 x, then 96 y ^ '… JEE Main 2024Let y = _ e ( 1 - x ^ 2 1 + x ^ 2 ), - 1 < x < 1 . Then at x = (1)/(2) , the value of… JEE Main 2024Suppose f(x) = ( 2 ^ x + 2 ^ - x ) x ^ - 1 ( x ^ 2 - x + 1) (7 x ^ 2 + 3x + 1) ^ 3 . Then… JEE Main 2024Let f ( x ) = x ^ 3 + x ^ 2 f ^ ' ( 1 ) + x f ^ " ( 2 ) + f ^ ''' ( 3 ) , x ∈ R . Then f… JEE Main 2024Let for a differentiable function f: ( 0, ∞ ) → R , f(x) - f(y) ≥ _ e ((x)/(y)) + x - y… JEE Main 2024 Full Differentiation list Differentiation JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs