# A Generalization of the Mean Value Theorem

A Generalization of the Mean Value Theorem

Theorem: Let be a function continuous on and differentiable on . Then there is a in such that .

f(x)

[a,b]

(a,b)

c

(a,b)

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 axis, the tangent line through , and the secant line through and the point is an isosceles triangle (the green triangle). Therefore the slopes of the two sides not on the axis are and .

x

(c,f(c))

(c,f(c))

(b,f(a))

x

f'(c)

-f'(c)

The example used is the function .

f(x)=+x++

c

0

c

1

c

2

2

x

c

3

3

x