## 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)

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Graeme McRae.