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Let f(x) is differentiable in(a,b) andf(x) not equal to zero for all X belongs to closed interval a to b Then there exists y belongs to (a, b) such that f'(y) /f(y)=

Let f(x) is differentiable in(a,b) andf(x) not equal to zero for all X belongs to closed interval a to b
Then there exists y belongs to (a, b)  such that 
f'(y) /f(y)=

Grade:12th pass

1 Answers

Aditya Gupta
2081 Points
3 years ago
consider the function g(x)= ln|f(x)|. obviously g(x) is defined everywhere in [a,b] as f(x) is not zero anywhere.
now, g(x) is differentiable everywhere in (a,b), with g’(x)= f’(x)/f(x).
applying LMVT on g(x), we have
g(b) – g(a)= (b – a)*g’(y) for some y lying in (a,b).
but g(b) – g(a)=  ln|f(b)| –  ln|f(a)|=  ln|f(b)/f(a)|=  (b – a)*f'(y) /f(y)
so f'(y) /f(y)= ln|f(b)/f(a)|/(b – a)

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