Let f:R be a twice-differentiable function such that (sin x cos y) [f(2x+2y)-f(2x-2y) ]=(cos x sin y) [f(2x+2y)+f(2x-2y) ] for all x,y. If f'(0)=1/2, then the value of 24 f'' ((5π)/3) is:

Limit, Continuity and Differentiability: JEE Main 2025 Mathematics question

JEE Main 2025 · April, Unspecified (session-level dataset) · Q29

JEE Main 2025April, Unspecified (session-level dataset)MathematicsMediumLimit, Continuity and Differentiability420× since 2002

Let

f:\mathbb{R}\to\mathbb{R}

be a twice-differentiable function such that

(\sin x\cos y)\bigl[f(2x+2y)-f(2x-2y)\bigr]=(\cos x\sin y)\bigl[f(2x+2y)+f(2x-2y)\bigr]

for all

x,y\in\mathbb{R}

. If

f'(0)=\tfrac12

, then the value of

24\,f''\bigl(\tfrac{5\pi}{3}\bigr)

is: