A Generalization of the Mean Value Theorem

c

0

0.5

c

1

0.1

c

2

1

c3

1

a

b

Theorem: Let

f(x)

be a function continuous on

[a,b]

and differentiable on

(a,b)

. Then there is a

c

in

(a,b)

such that

f'(c)=

f(c)-f(a)

b-c

.

Proof: the theorem follows by applying Rolle's theorem to the auxiliary function

h(x)=-(x-b)(f(x)-f(a)).

Here is a geometric interpretation: The triangle formed by the

x

axis, the tangent line through

(c,f(c))

, and the secant line through

(c,f(c))

and the point

(b,f(a))

is an isosceles triangle (the green triangle). Therefore the slopes of the two sides not on the

x

axis are

f'(c)

and

-f'(c)

.

The example used is the function

f(x)=

c

0

+

c

1

x+

c

2

x

+

c

3

x

.