Paragraph: Let f be a continuous function such that g(x)= _ -1 ^1 f(t)|x-t| d t where x (-1,1) . Question: The value of ₀^1 g^ (x) f(x) d x is equal to:

Step-by-step solution

For x (-1,1) g(x)= _ -1 ^x f(t)(x-t) d t+ _x^1 f(t)(t-x) d t=x _ -1 ^x f(t) d t- _ -1 ^x t f(t) d t+ _x^1 t f(t) d t-x _x^1 f(t) d t Now, differentiate aligned & g^ (x)= _ -1 ^x f(t) d t+x d d x _ -1 ^x f(t) d t- d d x _ -1 ^x t f(t) d t+ d d x _x^1 t f(t) d t- _x^1 f(t) d t-x d d x _x^1 f(t) d t & g^ (x)= _ -1 ^x f(t) d t+x f(x)-f(x) x-f(x) x-f(x) x- _x^1 f(t) d t+x f(x) & g^ (x)= _ -1 ^x f(t) d t- _x^1 f(t) d t .... (1) aligned Again differentiate g^ (x)= d d x _ -1 ^x f(t) d t- d d x _x^1 f(t) d t=f(x)+f(x)=2 f(