By definition of absolute value,
Differentiating once gives
As
from either side, we find that
, so really
Differentiating again, we find
but this time, the limit does not exist as
from either side, so
is undefined. However, we see that
when
, and
when
, and we know that
is continuous at
. This means the concavity must change at
, so (0, 0) is the inflection point. (The takeaway is that inflection points *can* occur when the second derivative is undefined, but not always.)