Understanding the Mean value theorem

[chwala]

Homework Statement

Can we have two tangents (two turning points) within the given two end points just asking? I know the theorem holds when there is a tangent to a point $c$ and a secant line joining the two end points.

Relevant Equations

Understanding of;
-Mean Value THeorem
-Rolle's Theorem $(f(a)=f(b)$ and one tangent line only...
...extended mean value theorem

Can we have two tangents (two turning points) within the given two end points just asking? I know the theorem holds when there is a tangent to a point $c$ and a secant line joining the two end points.
Or Theorem only holds for one tangent point. Cheers

 

[chwala]

I just checked Wikipedia...yep its possible to have two tangents parallel to the secant...phew a lot of things to read and re-familiarize!

 

[fresh_42]

Consider the sine function and take start and end points far enough away from each other.

 

[chwala]

Consider the sine function and take start and end points far enough away from each other.

True, I see...then in this case we have only two tangent lines... touching an infinite number of turning points...

 

[fresh_42]

True, I see...then in this case we have only two tangent lines... touching an infinite number of turning points...

We could also consider all tangents at $x=(2k+1/4)\pi.$ (MVT)

 

[fresh_42]

Here is a nice application of the MVT:

5. Is there always a position on a floor (continuous, no holes, steps etc.) for a rectangular table with four equal legs, such that the table does not wiggle?
https://www.physicsforums.com/threads/math-challenge-december-2018.961292/