Your textbook needed an editor who could understand that the punctuation of the definition was rotten. It's be better to say
The point $x = c$ of a curve $y = f(x)$ defined by a function $f$ that's twice differentiable almost everywhere is an inflection point if either
(a) $f''(c) = 0$ and $f'$ changes sign as $x$ increases through $c$, or
(b) $f''(c)$ is undefined and $f'$ changes sign as $x$ increases through $c$.
The final clause ("The latter condition") is just plain wrong, as the example $y = x^4$ shows, because although $f''(0) = 0$, the concavity of the function is "up" on both sides of $x = 0$.
It also doesn't handle cases like $y = x^4 \sin (1/x)$ for $x \ne 0$ and $ y = 0$ for $x = 0$, which have second derivative zero, but for which the curvature changes sign infinitely often in any neighborhood of the origin -- one might call that an inflection point, or might not (I'd say "not", given the choice), but the authors' "as $x$ increases through $c$" suggests that they expect $f''$ to have one sign to the left of $c$ and the opposite sign to the right of $c$, at least locally, and for this function, that's just not true.
No.
Consider $f(x) = x^4$.
In order for there to be a change in concavity there needs to be a change of sign for the second derivative around our potential inflection point. However, $f''(x)=12x^2$ is positive to both sides of $x=0$ so it's not an inflection point.
You can visually tell this isn't an inflection point because the graph looks like a steeper parabola.
Best Answer
You can think about it this way, at every point of inflection $f''(x)=0$ or is $undefined$ but the the function $f(x)$ must change from concave upwards to downwards( the sign of $f''(x)$ must change). that being said, for every $x$ value where $f''(x)=0$ or is $undefined$ are only $possible$ points​ of inflection , you have to check weather the sign of $f''(x)$ changes or not.
In short, if $(c,f(c))$ is a point of inflection then we can say:
The sign of $f''(x)$ changes at $x=c$
$f''(c)$ is either 0 or undefined