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I refered these questions Definition of Point of Inflection An inflection point where the second derivative doesn't exist? What is inflection point? I couldnt understand the necessary conditions for point of inflection.My specific doubt is Can point of inflection occur at a point where the second derivative of a function is discontinuous? In the image attached, the curves of the actual function, its second derivative and third derivative are given(from left to right correspondingly) Can the point x=b be an inflection point for the function.enter image description here

Mohan
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  • You have answered your own question. https://math.stackexchange.com/questions/402459/an-inflection-point-where-the-second-derivative-doesnt-exist Yes a point where the second derivative is discontinuous at that point can be an inflection point. – Matthew Liu Feb 01 '19 at 07:34
  • @Matthew Liu Ok.Thanks for answering.Does this mean the point x=b in the diagram can be a point of inflection for f(x) ? – Mohan Feb 01 '19 at 07:43
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    I think it "point of inflection" is only defined as change in sign of the second derivative and that this change of sign does not need to be continuous and go through 0. But it was ages ago I even saw this expression. – mathreadler Feb 01 '19 at 08:06
  • @mathreadler A point of inflection is a geometric property. It's a point where the tangent line crosses the graph. – B. Goddard Feb 01 '19 at 11:39
  • @B.Goddard Hmm if you say so. I think last time I encountered the name was in high school... Definitely did not occur particularly often at university for me. – mathreadler Feb 01 '19 at 13:12

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The short answer is "yes." An inflection point is a point where the tangent line crosses the graph of the function. So if the tangent line is vertical at a point, then the first derivative doesn't exist and therefore the second doesn't exist.

Take $y=\sqrt[3]{x}$. Then $(0,0)$ is an inflection point, but the second derivative is discontinuous there.

B. Goddard
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