Mean Value Theorem For Derivatives, Definition, Example, Proof

The Mean Value Theorem for derivatives illustrates that the actual slope equals the average slope at some point in the closed interval.

Mean Value Theorem For Derivatives, Definition, Proof

Statement
The theorem states that, If f be a differentiable on an open interval (a,b) and continuous on a closed interval [a,b] ,where a < b then there is c in (a,b) such that, f '(c) = (f(b) - f(a)) / (b - a)

Proof
Let g(x) = (f(b) - f(a)) x - (b-a) f(x)
Then g(x) is a continuous function on [a,b] and differentiable in (a,b).
First Order derivative of g(x) is,
g'(x) = (f(b) - f(a)) - (b-a) f '(x)

if a < x < b then g'(x) = 0 by Rolle's Theorem
0 = (f(b) - f(a)) - (b-a) f '(x)
(b-a) f '(x) = (f(b) - f(a))
f '(x) = (f(b) - f(a)) / (b-a)

Note
g(x) = (f(b) - f(a)) x - (b-a) f(x)

let x = a,
g(a) = (f(b) - f(a)) a - (b-a) f(a)
= f(b).a - f(a).a - b.f(a) + a.f(a)
= a.f(b) - b.f(a)

let x = b,
g(b) = (f(b) - f(a)) b - (b-a) f(b)
= f(b).b - f(a).b - b.f(b) + a.f(b)
= a.f(b) - b.f(a)

g(a) = g(b)

By Rolle's Theorem, g'(x) = 0 is proved.

Hence the Mean Value Theorem for Derivatives is proved.
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