The Mean Value Theorem and Its Consequences
Rolle's Theorem
Suppose we pick any two points on the -axis and think about all of the differentiable functions which go through those two points (Fig. 1).
Fig. 1
Since our functions are differentiable, they must be continuous and their graphs can not have any holes or breaks. Also, since these functions are differentiable, their derivatives are defined everywhere between our two points and their graphs can not have any "corners" or vertical tangents. The graphs of the functions in Fig. 1 can still have all sorts of shapes, and it may seem unlikely that they have any common properties other than the ones we have stated, but Michel Rolle found one. He noticed that every one of these functions has one or more points where the tangent line is horizontal (Fig. 2), and this result is named after him.
Fig. 2
Rolle's Theorem: If , and is continuous for and differentiable for ,
then there is at least one number , between and , so that .
Proof: We consider two cases: when for all in and when for some in .
Case I, for all in : If for all between and , then
is a horizontal line segment and for all values of strictly between and .
Case II, for some in : Since is continuous on the closed interval , we know from the Extreme Value Theorem that must have a maximum value in the closed interval and a minimum value in the interval.
If for some value of in , then the maximum of must occur at some value strictly between and . (Why can't the maximum be at or ?) Since is a local maximum of ,
then is a critical number of and or is undefined. But is differentiable at all between and , so the only possibility left is that .
If for some value of in , then has a minimum at some value strictly between a and , and .
In either case, there is at least one value of between and so that .
Example 1: Show that satisfies the hypotheses of Rolle's Theorem on the interval and find the value of which the theorem says exists.
Solution: is a polynomial so it is continuous and differentiable everywhere. and . so at and .
The value is between
and . Fig. 3 shows the graph of .
Fig. 3
Practice 1: Find the value(s) of c for Rolle's Theorem for the functions in Fig. 4.
Fig. 4