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