Two tangent at the same point of a function

[chrisgk]

it is possible to have two different tangents at the same point of a function?

 

[MarkFL]

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:

\(\displaystyle f(x)=|x|\)

Now, the family of lines given by:

\(\displaystyle y=mx\) 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]

 

[chrisgk]

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?

 

[MarkFL]

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.