For y = f(x), the second derivative of f, provided that it exists, is:
f''(x) = d/dx f'(x)
Other notations for f''(x) are:
d2y/dx2 and y''
For the interval (a, b), if f'' > 0, then f is concave upward, and if f'' < 0, then f is concave downward.
An inflection point is a point on the graph where the concavity changes from upward to downward or downward to upward.
If (c, f(c)) is an inflection point of f, then c is a partition number for f''.
+ve | -ve | 0 | Not defined | |
---|---|---|---|---|
f(x) | Above x-axis | Below x-axis | x-intercept | Vertical asymptote? |
f'(x) | Increasing | Decreasing | Local extremum? | Local extremum? |
f''(x) | Concave upwards | Concave downwards | Inflection point? | Inflection point? |