Rolle's Theorem
Rolle's theorem is one of the foundational theorems in differential calculus. It is a special case of, and in fact is equivalent to, the mean value theorem, which in turn is an essential ingredient in the proof of the fundamental theorem of calculus.
Contents
Summary
The theorem states as follows:
Rolle's Theorem
For any function that is continuous within the interval and differentiable within the interval where there exists at least one point where within the interval
A graphical demonstration of this will help our understanding; actually, you'll feel that it's very apparent:
Imgur
In the figure above, we can set any two points as and as long as and the function is differentiable within the interval Then, of course, there has to be a point in between where which is the red point in the diagram. Now let's take a look at the mathematical proof of this theorem.
We divide it into two cases:
(1) is a constant function.
If is a constant function, then for the whole interval. Then, of course, there exists a such that within the interval
(2) is not a constant function.
When is not a constant function but is continuous within the interval according to the extreme value theorem, must have a maximum function value and minimum function value within the interval Since is not a constant function, at least one of the extrema must exist within the interval
(2)-1
If has its maximum function value at then for a real number whose absolute value is small enough that it follows that
Hence we have
Since is differentiable in the interval according to the squeeze theorem we have
(2)-2
If has its minimum function value at then for a real number whose absolute value is small enough that it follows that
Hence we have
Since is differentiable in the interval according to the squeeze theorem we have
Therefore, whichever case we are given, there exists a point where within the interval
Obviously, for Rolle's theorem to hold, the function must be differentiable within the interval we are considering. Thus Rolle's theorem cannot be applied to functions like
Example Problems
When
show that has at least one root in the interval using Rolle's theorem.
Observe that is continuous in the interval and differentiable in
The function values of at are
Then from and , it is confirmed that Rolle's theorem can be applied. According to Rolle's theorem, there exists a point where in the interval
When
show that has at least one root in the interval using Rolle's theorem.
Observe that is continuous in the interval and differentiable in
The function values of at are
Then from and , it is confirmed that Rolle's theorem can be applied. According to Rolle's theorem, there exists a point where in the interval
Show that the following formula has at least one root in the interval
Let Then
where is the constant of integration. Since is continuous in the interval and differentiable in the interval and we can apply Rolle's theorem. According to Rolle's theorem, there must be a point where within the interval
Therefore the given equation has at least one root in the interval
Show that the following formula has at least one root in the interval
Since is never equal to zero within the given interval, the given equation is equivalent to
Let Then
where is the constant of integration. Since is continuous in the interval and differentiable in the interval and we can apply Rolle's theorem. According to Rolle's theorem, there must be a point where within the interval
Therefore the given equation has at least one root in the interval