Inflection if y'' = 0 and y'' changes sign - ie. change in concavity as Xayma said.
Horizontal inflection if y'' = 0 and y' = 0 and y'' changes sign - be carfeful with this one, you still need the change in sign. For example, y = x^3 has a horizontal poi at the origin, y = x^4 has a minimum tp at the origin, but both have y'' = 0 and y' = 0.
Vertical inflection is more difficult to demonstrate - my first attempt at writing a definition had a few holes (like for absolute value graphs), so after thinking, here is my approach:
- There must be a vertical tangent at the point in question.
- y'' must change change sign
- y'' will usually be undefined - I can't think of a case off hand where y'' = 0, but I won't exclude the possibility without further thought. (Anyone else have a view on this?). y'' certainly CANNOT be anything else (ie. couldn't have y'' = 5 (say), but then that is obvious as this would prevent y'' changing sign, so long as y'' is continuous).
Now, there will be a vertical tangent at some point x = a if y is defined when x = a, y' is undefined when x = a, and the limits as x approaches a from BOTH above and below show that y' goes to positive or negative infinity.
In the above example, y = x^1/3, we know y' = x^(-2/3) / 3.
At (0, 0), y' is undefined (its 1 / (3 * 0).
As x ---> 0+, x^2/3 ---> 0+, so y' ---> 1 / (3 * 0+) ---> + inf
As x ---> 0-, x^2/3 ---> 0-, so y' ---> 1 / (3 * 0-) ---> - inf
Thus, we have a vertical tangent at (0, 0).
Further, y'' is undefined, but also changes sign at (0, 0). Thus my conclusion that (0, 0) is a vertical point of inflection.
(Note: there is a much easier way to do this. As Xyama said above, this graph is the reflection of y = x^3 in the line y = x. That is, it is the inverse function of y = x^3. Thus, the horizontal poi at (0, 0) of y = x^3 becomes a vertical poi of y = x^1/3.)
To recognise the importance of there being a vertical tangent, consider the graph of y = |x^2 - 1|. y'' is undefined at (1, 0), and does change signs. y' is also undefined at (1, 0). However, there is no vertical tangent (in fact, there is no tangent at all) because as x approaches 1 from above, y' approaches 2, and as x approaches 1 from below, y' approaches -2. Thus (1, 0) is not a point of inflection.
