Stationary points of inflection & points of inflection

[Henreezy]

I'm kind of confused about analysing behaviour using second derivatives.. I know when it's a max or a min but I'm unsure what they mean when they say look for a sign change to see whether it is St. point of inflection or a point of inflection. Can somebody help me out?

[leflyi]

Look at the graph of

Implies a gradient of 0, at x = 0.
  Then f(x) implies that the turning point is a local minimum. as

In turn

Implies a gradient of 0, at x = 0.
  Then f(x) implies that the turning point is a local max as.

Now lets look at

Implies turning point of x=0
Then f''(x) implies the turning point is in fact a stationary point, as f''(x) = 0, when x=0.

That is basically the behaviour of second derivatives..

Jut sub in the value of dy/dx for the gradient being 0, into x for d2y/dx2 and if the value is less that 0, that is a maximum, more the 0, minimum, or equal to zero POI; you can prove it through a graph to visualize.

[Henreezy]

I understand those parts but I'm unsure where you make the distinction between stationary points of inflection and a regular point of inflection.

[lzxnl]

A regular point of inflection of y=f(x) is a point (a, f(a)) such that x=a is an extrema of f'(x); f''(x) changes sign at x=a. If you look at f(x)=x^3. you'll see what I mean. f''(x)=6x and changes sign at the origin.
The reason why this sign change is necessary is apparent if we look at gradient curves. Let us assume that to the left of x=a, f''(x)<0 and to the right, f''(x)>0
Then, to the left of x=a, the gradient is decreasing. To the right, the gradient is increasing. Therefore, x=a is a local minimum of y=f'(x).
If you see points of inflection as points of maximum/minimum gradient, you'll see why this is necessary. Only at a stationary point of y=f'(x) will you therefore have a point of inflection. However, the sign change is necessary to ensure that you have a local max/minimum at x=a which is necessary for a point of inflection of f(x).
Otherwise, you'll have what happens with x^4.
Apologies if my explanation is a bit roundabout.