Does the concavity of a twice differentiable change every time its double differetiation is zero?

According to me it should not always be the case as a linear function in x ha double derivative zero at every point but does not change concavity.

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1

 

‘According to me’. I’m going to find a way to put that in a paper.
– Mattos
2 days ago

2

 

Convexity changes iff the second derivative changes sign. Just to become zero it is not enough. Example: f(x)=x4f(x)=x^4.
– A.G.
2 days ago

  

 

What about f(x)=0f(x) = 0?
– Ilya
2 days ago

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