Rolle's theorem
In
calculus, Rolle's theorem states that if a function
f is continuous on a closed interval and
differentiable on the open interval , and then there is some number
c in the open interval such that
Intuitively, this means that if a smooth curve is equal at two points then there must be a stationary point somewhere between them. All the assumptions are necessary. For example, if
the
absolute value of
x, then we have that
but there is no
x between -1 and 1 for which . This is because that function, although continuous, is not differentiable at
x=0.
Encyclopedia
In
calculus,
Rolle's theorem states that if a function
f is continuous on a closed interval and
differentiable on the open interval , and then there is some number
c in the open interval such that
- .
Intuitively, this means that if a smooth curve is equal at two points then there must be a stationary point somewhere between them. All the assumptions are necessary. For example, if
the
absolute value of
x, then we have that
-
but there is no
x between -1 and 1 for which . This is because that function, although continuous, is not differentiable at
x=0.
The statement of the theorem was first elucidated by
Indian astronomer Bhaskara in the 12th century. The theorem was reproduced centuries later by Michel Rolle in 1691.
Rolle's Theorem is used in proving the
mean value theorem, which eliminates the requirement that .
Proof
The idea of the proof is to argue that if then f must attain either a maximum or a minimum somewhere between and , and at either of these points.
Now, by assumption, is continuous on , and by the continuity property attains both its maximum and its minimum in . If these are both attained at endpoints of then is constant on and so at every point of .
Suppose then that the maximum is obtained at an interior point . We wish to show that . We shall examine the left-hand and right-hand derivatives separately.
For less than , is non-negative, since is a maximum. Thus the limit is non-negative. .
For greater than , is non-positive. Thus is non-positive.
Finally, since is differentiable at , these two limits must be equal and hence are both 0. This implies that .
Relaxed assumputions
The theorem is usually stated in the form above, but it is actually valid in a slightly more general setting:
We only need to assume that
-
is continuous on , that
-
and that
-
Generalization
Suppose a function
f has
n derivatives, , and
f has
n+1 roots, for distinct points , in . Then there is such that .
See also
External links
