Two tangent at the same point of a function

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Discussion Overview

The discussion revolves around the possibility of having two different tangent lines at the same point of a function, particularly focusing on the conditions under which this may occur, such as differentiability and convexity.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions whether it is possible to have two different tangents at the same point of a function.
  • Another participant asserts that if a function is differentiable at a point, it can only have one tangent line, but if it is convex and not differentiable, multiple tangent lines may exist, citing the example of the function \( f(x) = |x| \).
  • A participant reiterates the question about having a tangent at a point where the function is not differentiable.
  • Further clarification is provided that if a tangent line is defined as merely touching the function without crossing it, then multiple tangent lines can exist at non-differentiable points, as illustrated by the example of \( f(x) = |x| \). However, if a tangent line is required to have the same slope as the function at that point, then differentiability is necessary for a unique tangent line.

Areas of Agreement / Disagreement

Participants express differing views on the definition of a tangent line and the conditions under which multiple tangents can exist, indicating that the discussion remains unresolved.

Contextual Notes

Participants have not reached a consensus on the definitions of tangent lines and the implications of differentiability on the existence of multiple tangents.

chrisgk
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it is possible to have two different tangents at the same point of a function?
 
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If a function is differentiable at a given point, then it can only have one tangent line, but if a function is convex and not differentiable at that point, then it can have multiple tangent lines. For example, consider the function:

$$f(x)=|x|$$

Now, the family of lines given by:

$$y=mx$$ where $-1<m<1$

touch $f$ only at the origin. :D

[DESMOS=-10,10,-3.469210754553339,3.469210754553339]y=\left|x\right|;y=mx;m=0.5[/DESMOS]
 
we can have tangent at a point that function is not differentable?

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we can have tangent at a point that function is not differentiable?
 
chrisgk said:
we can have tangent at a point that function is not differentable?

If we simply define a tangent line as a line that touches a function at a given point without crossing over the curve, then the example I gave shows that the function $f(x)=|x|$ has no unique tangent line at the origin.

However, if we require a tangent line to have the same slope that the function has at the tangent point, then we require the function to be differentiable at that point, and there will be a unique tangent line.
 

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