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

Step-by-step solution

aligned & ( x y)(f(2 x+2 y)-f(2 x-2 y))=( x y) & (f(2 x+2 y)+f(2 x-2 y)) & f(2 x+2 y)( (x-y))=f(2 x-2 y) (x+y) & f(2 x+2 y) (x+y) = f(2 x-2 y) (x-y) & Put 2 x+2 y=m, 2 x-2 y=n & f(m) ( m 2 ) = f(n) ( n 2 ) = K & f(m)=K ( m 2 ) & f(x)=K ( x 2 ) & f^ (x)= K 2 ( x 2 ) & P_ Put x=0 ; 1 2 = K 2 K=1 & f^ (x)= 1 2 x 2 aligned aligned & f^ (x)=- 1 4 x 2 & 4 f^ ( 5 3 )= (- 1 4 ( 5 6 ) ) 24 & = -24 8 =-3 aligned