Doubts about inflection point If the first derivative of a function at a point is either zero or undefined and the sign of the first derivative around that point doesn't change, then can we say that that point is the inflection point? Or, do we necessarily have to check if the second derivative is zero there?
Also, if the first derivative doesn't exist, how can we comment about second derivative?
If we have $$f(x)=\begin{cases}x^2;&x\le2\\4;&x\ge2\end{cases}$$
Here, what can we comment about $x=2$? Is it an inflection point?
 A: 
If the first derivative of a function at a point is either zero or undefined and the sign of the first derivative around that point doesn't change, then can we say that that point is the inflection point?

Unfortunately not. The first derivative of $f(x)=3$ is zero at, and its sign technically doesn't change around, $x=7.$

Or, do we necessarily have to check if the second derivative is zero there?

$f(x)=x^{\frac13}$ has no second derivative, but is an inflection point, at $x=0.$

Also, if the first derivative doesn't exist, how can we comment about second derivative?

If the curve's second derivative changes sign around a point, then the point is an inflection point regardless of whether its derivative exists. The previous example illustrates this.

If we have $$f(x)=\begin{cases}x^2;&x\le2\\4;&x\ge2\end{cases}$$
Here, what can we comment about $x=2$? Is it an inflection point?

Since the second derivative doesn't change sign around that point, it isn't an inflection point.
The key point is that an inflection point is one where around it, the second derivative changes sign. (Since curvature and second derivative have the same sign, this is equivalent to saying that the curvature changes sign.)
