⋆ Pattern – 25 (Mean value theorem)

If f is a continuous function on [a, b], then there exists c ∈ (a, b) such that

f(c)  = 

Proof :

Let, F(x) = —-(1) x ∈ [a, b]

It is obvious that F(x) is differentiable function on (a, b) also, f is continuous function on [a, b]

∴  F(x) is continuous function on [a, b]

Hence,

F is continuous on [a, b] & differentiable on (a, b), then there exists c ∈ (a, b) such that F’(c) = ——(2)

Now, from (1)

F’(x) = f(x) × 1 – f(a) × 0

= f(x) ——(3)

∴  From 2 & 3,

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