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`        this question i think is based onon some theorem.. Please answer it methodically..`
one year ago

```							 this ques merely looks difficult but is easy, and is a direct application of LAGRANGES MEAN VALUR THEOREM.let mu=u and lambda=mnote 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 2bf’(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 thatLHS=(b-a)[∫f”(x)dx from m to u+uf”(u)]
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one year ago
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