this ques merely looks difficult but is easy, and is a direct application of LAGRANGES MEAN VALUR THEOREM.
let mu=u and lambda=m
note that ∫xf”(x)dx= xf’(x) – f(x) [integration by parts]
so ∫xf”(x)dx from a to b= bf’(b) – f(b) – [af’(a) – f(a)] = bf’(b) – af’(a)+f(a) – f(b) =LHS.......(1)
Now consider the following functions:
g(x)= xf’(x) and h(x)= f(x) [note that g’(x)= f’(x)+xf”(x) and h’(x)= f’(x)]
applying LMVT on both we have
[g(b)–g(a)]/(b-a)=g’(u) for some u in (a,b) and [h(b)–h(a)]/(b-a)=h’(m) for some m in (a,b)
bf’(b)–af’(a)=(b-a)[f’(u)+uf”(u)].....(2)
and f(b)-f(a)=(b-a)f’(m).....(3)
subtracting 3 from 2
bf’(b)–af’(a) – f(b)+f(a)= (b-a)[f’(u)+uf”(u) – f’(m)]
from 1,
LHS=(b-a)[f’(u)+uf”(u) – f’(m)]
note that ∫f”(x)dx from m to u= f’(u) – f’(m)
so that
LHS=(b-a)[∫f”(x)dx from m to u+uf”(u)]
