Given that d d x ₀^ (x) f(t) d t=f( (x)) ^ (x) . For all x (0, 2 ) , if ₁^ x t^2 f(t) d t= 2 x , then f ( 1 2 )=

Step-by-step solution

aligned & ₁^ x t^2 f(t) d t- 2 x & ( x)^2 f( x) ( d d x x )=-2 2 x & ^2 x f( x)(- x)=-2 2 x & f( x)= -2 2 x - ^2 x x & At x= 4 ; f ( 4 )= -2 2 - ^2 4 4 & f ( 1 2 )= 2 1 1 2 1 2 f ( 1 2 )=4 2 aligned