Continuity and Differentiability
Non-differentiability of maximum powers
nta_pyq_2025_apr
Grade 12
Question:
Let f : R$\to$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))$, for all x, y$\ in $R. If f (0) = ′ 1 2 , then the value of 24f ′′ ( 5$\pi$3 ) is:
Step-by-Step Solution
Key Concept: Break the function at its$formula-changing$points and compare$one-sided$limits or derivatives there.
(sin x cos y)(f$(2x + 2y) - f$$(2x - 2y)) = (cos$x sin y) (2) (f$(2x + 2y) + f$$(2x - 2y))$f$(2x + 2y)(sin(x - y)) = f$$(2x - 2y)$$sin(x + y)$f$(2x + 2y)$f$(2x - 2y) = sin(x + y)$$sin(x - y)$Put$2x + 2y = m$,$2x - 2y = n$f (m) f (n) = = K m n sin( ) sin( ) 2 2 m $\Rightarrow$ f$(m) = K$sin( ) 2 x ∴ f$(x) = K$sin( ) 2 K x ′ f$(x) = cos($) 2 2 1 K PPut$x = 0$; = $\Rightarrow$$K = 1$2 2 1 x ′ f$(x) = cos$2 2 1 x ′′ f (x) = - sin 4 2 5$\pi$1 5$\pi$′′ 4f ($) = (- sin($)) 24 3 4$6 -24$= = -3 8
Correct Answer: 3