IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

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Let a function g be defined as :
g(x) = (x-a)(x-b)f(x)

Since f(x) is continuous in [a,b] and differentiable in (a,b) , hence g(x) is also continuous in [a,b] and differentiable in (a,b).

Now g(a) = 0 and g(b) = 0

Hence Rolle's theorem is applicable to g(x) in (a,b).
There exists at least one c in (a,b) for which g'(c) = 0

g'(x) = (x-a)f(x) + (x-b)f(x) + (x-a)(x-b)f'(x)
g'(c) = (c-a)f(c) + (c-b)f(c) + (c-a)(c-b)f'(c) = 0

Divide the whole equation be (c-a)(c-b)f(c) :
1/(c-b) + 1/(c-a) + f'(c)/f(c) = 0

f'(c)/f(c) = 1/(a-c) + 1/(b-c)

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