Understanding the Mean value theorem

chwala
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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
 
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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!
 
Consider the sine function and take start and end points far enough away from each other.
 
fresh_42 said:
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...
 
chwala said:
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)
 

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