$1.$ If $\,f'(a) \neq 0\, \text{ and } \,f''(a) = 0\,$, then there is a non-stationary point of inflection at $\,x = a\,$.To confirm, you check the sign of $\,f''(x)\,$ for values of $x$ on the left and right of $a$. If the sign changes, that means concavity changes, and thus, $a$ is the non-stationary point of inflection.
$2.$ If $\,f'(a) = 0\, \text{ and } \,f''(a) = 0\,$, then there is a stationary point of inflection at $\,x = a\,$. To confirm, you check the sign of $\,f''(x)\,$ for values of $x$ on the left and right of $a$. If the sign changes, that means concavity changes, and thus, $a$ is the stationary point of inflection.
Confusion 1: Are $1$ and $2$ correct statements?
Confusion 2: It's said that if $\,f''(a) = 0\, \text{ and } \,f'''(a) \neq 0\, (i.e., \,f'''(a)\,$ is actually some well-defined number and that number is not $0$), then there is an inflection point at $x=a$. My dilemma is that shouldn't it be as $\,f'''(a) =0 \,$ and if there is a change of sign in $f'''(x)$ for values of $x$ on the left and right of $a$, then $x=a$ can be termed as the point of inflection? Also, will this inflection point always be non-stationary?
Additionally, I would appreciate it if you could demonstrate the $f'(x)$, $f''(x)$, and $f'''(x)$ at $x=a$ through graphs and then define the point of inflection.
I have looked at the comments and since the length of my response doesn't fit the comments, I have put it below:
As $f ' ' '(a)≠0$ so $f ' ' '(a)>0$ or $f ' ' '(a)<0$ and that in turn means $f ' '(a)>0$ or $f ' '(a) <0$ around the point $x=a$. This means $f' '$ is strictly monotonic but it is either positive on both sides of $x=a$ or negative on both sides of $x=a$. There will be no change in the sign of $f ' '$ but concavity will change at $x=a$. That is, if you consider $f $ as the function then slope of $f $ will be $f '$ and at point of inflection $f ' '$ will be equal to 0. And in the immediate neighbourhood of the inflection point(both sides), slope ( in this case $f ' $) will either go from less increasing to more increasing OR less decreasing to more decreasing and thus the change in concavity. But it will never change sign. Now consider $f '$ as the function and its slope as $f ' '$ and then $ f ' ' '$ as the requirement for point of inflection. Following what is stated above, if $x=a$ is an inflection point then change in slope (i.e. $ f ' ' '$) will be zero at $x=a$ but it will either be positive on both sides of $a$ or negative on both of $a$ i.e slope $ f ' '$ will either go from less increasing to more increasing OR less decreasing to more decreasing.
So, shouldn't $f ' ' '(a)=0$ if $x=a$ is an inflection point rather than $f'(a) \neq 0$? I am still confused and look forward to hearing from you all. I hope I have been clear in putting up my dilemma.
"Stationary" means $f'(a) = 0$. Nothing more or less.
An inflection point is a point where the concavity of the function changes direction. For twice differentiable functions, $f'' > 0$ where the function is concave up, and $f'' < 0$ where the function is concave down. $f''$ can only switch from one to the other by passing through $0$ (even without assuming $f''$ is continuous, because it is a derivative). Thus at an inflection point $a, f''(a) = 0$.
But it is possible for $f''(a) = 0$ without $a$ being an inflection point. $y = f''(x)$ could touch the line $y = 0$, but then turn around and leave in the same direction from which it came, like $y = x^2$ does. But that would mean that $a$ was a local extremum of $f''$, and therefore that the derivative of $f''$ is $0$. I.e., if $f''(a) = 0$, but $a$ is not an inflection point, then $f'''(a) = 0$ as well.
Thus if $f'''(a) \ne 0$, we know this doesn't occur, and thus we know that $a$ must be an inflection point even though we haven't directly checked that $f''$ has opposite signs on either side of $a$.
So, to find where a thrice-differentiable $f$ has inflection points, you find each of the points $a$ where $f''(a) = 0$, as these are the candidate values where inflection points might occur. To determine if they actually are inflection points, you have two choices: