Understanding the Mean value theorem

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The discussion centers on the Mean Value Theorem (MVT) and its implications regarding tangent lines and turning points. It confirms that multiple tangent lines can exist between two endpoints, particularly when considering functions like sine over a sufficiently large interval. The conversation highlights that while the MVT guarantees at least one tangent, it can accommodate multiple tangents touching numerous turning points. Additionally, the concept of finding a stable position for a rectangular table on a continuous floor is presented as a practical application of the MVT. Overall, the thread emphasizes the flexibility of the MVT in various mathematical contexts.
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)
 

Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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