Flett's Theorem

a

0

a

1

a

3

a

Flett's theorem: Given a function

f(x)

differentiable on

[a,b]

with

f'(a)=f'(b)

, there is an intermediate point

c

such that

f'(c)=

f(c)-f(a)

c-a

.

Geometric interpretation: The secant line connecting the points

(a,f(a))

and

(c,f(c))

is exactly the tangent line to the curve

y=f(x)

at the point

(c,f(c))

.

The example used is the function

f(x)=

a

0

+

a

1

x+

a

3

x

on the interval

[-a,a]

.