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A direct extract from my book which states.(I have attached a photo as well)

"A point P(Xo,Yo) on the curve of a function y=f(x) is a point of inflection if f"(Xo)=0 and either f"(x) changes its sign at X=Xo, or [third derivative] i.e. f"'(Xo)≠0."

We can find pretty many examples for point of inflection where f"(Xo)=0 and f"(X) changes its sign at X=Xo. Like Y=X^3 at X=0. f"(X) is negative for (-∞,0) and positive for (0, +∞). So 0 is a point of inflection.

But i am really having hard time finding an example for second condition

From book

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  • $\begingroup$ If all derivatives of $f$ are continuous, then $f''(x_0) = 0$ and $f'''(x_0) \neq 0$ implies that $f''(x)$ changes sign at $x_0$. $\endgroup$
    – Arthur
    Nov 19 '16 at 11:22
  • $\begingroup$ Can you be more specific with an example? And I guess, according to second condition f"(x) should not change sign...but can be 0 and third derivative not equal to 0 at inflection point. $\endgroup$
    – Sahil
    Nov 19 '16 at 11:30
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By "second condition", do you mean

If $f''(x_{0})=0$ and $f'''(x_{0})\neq0$?

It implies the first condition. If $f''(x_{0})=0$ and $f'''(x_{0})\neq0$ then $f''(x)$ has different signs on either side of $x_{0}$. This is because $f'''(x_{0})$ is the slope of the tangent line to $f''(x_{0})$ at $x_{0}$.

The example you gave can still be used: $f(x)=x^{3}$, therefore $f''(x)=6x$ and $f'''(x)=6$. At $x_{0}=0$, $f''(0)=0$ and $f'''(0)=6$, and $0$ is an inflection point.

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    $\begingroup$ Note that we can have $f'''(x_0) = 0$ and still $f''(x)$ changes sign. For instance, $f(x) = x^5$, where $f''(x) = 20x^3$ changes sign but $f'''(x_0) = 60\cdot 0^2 = 0$. $\endgroup$
    – Arthur
    Nov 19 '16 at 11:38
  • $\begingroup$ Ah yes. I should change it to the second condition implies the first? $\endgroup$ Nov 19 '16 at 11:44
  • $\begingroup$ I got it. Cheers to all. It meant the same thing by applying "or" in between statements. I took it the other way(wrong way). Finally, I can conclude no inflection can occur without change in the sign of derivative. $\endgroup$
    – Sahil
    Nov 19 '16 at 12:23

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