If f(x) and g(x) are continuous functions in [a, b] and they are differentiable in (a, b), then prove that there exists c ∈ (a, b) such that
= (b –a) 
Text Solution
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Sol. Consider the function φ (x) = f(a) g(x) – f(x) g(a) for all x ∈ [a, b]
As f(x) and g(x) are continuous on [a, b] and differentiable on (a, b) therefore φ (x) is continuous on [a, b] and differentiable on (a, b). Consequently, there exists c ∈ (a, b) such that
φ′ = 
Now,
φ (x) = f(a) g(x) – f(x) g(a)
⇒ φ′ (x) = f(a) g ′ (x) –f ′ (x) g(a)
⇒ φ′ = f(a) g ′ –f ′ g(a) = 
Also,
φ = f(a) g(b) – f(b) g(a) = 
and,
φ = f(a) g(a) –f(a) g(a) = 0
∴ φ′ = 
⇒ 
= 
⇒ 
= (b –a) 
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