Mean Value Theorem

Let f be continuous on [a,b] and differentiable on (a,b).
Then there exists a number, c, in (a,b) such that f'(c) = (f(b)-f(a)) / (b-a)

Proof:

Let m = (f(b)-f(a)) / (b-a), the slope of the secant line that passes through points (a,f(a)) and (b,f(b))

Let g(x) = f(x) - m(x-a).

Then g(a) = f(a) and g(b) = f(b) - (f(b)-f(a))(b-a)/(b-a) = f(a),
so g(a) = g(b) = f(a)

By Rolle's Theorem, there exists a number, c, in (a,b) such that g'(c) = 0.

g'(x) = f'(x) - m, so
f'(x) = g'(x) + m, so
f'(c) = g'(c) + m, so
f'(c) = m, proving the theorem.

How is this theorem used?

The Mean Value Theorem is used to prove the Fundamental Theorem of Calculus.

It also figures in the proof of irrationality of Pi.

Related Pages in this Website

Rolle's Theorem -- that if f(a)=f(b) then f'(c)=0 for some c in (a,b)

Fundamental Theorem of Calculus -- if f is the derivative of F, then the integral from a to b of f(x)dx is F(b)-F(a)

The webmaster and author of the Math Help site is Graeme McRae.
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