Definition: An inflection point is a point on a continuous graph where the concavity changes, so the second derivative changes sign there.
What is an inflection point?
It is where the curve switches from bending up to bending down, or the reverse. At such a point, f” is zero or undefined and changes sign.
How do you find one?
- Compute f”(x).
- Solve f”(x) = 0 or find where it is undefined.
- Check that the sign actually changes.
Worked example: g(x) = x4 – 6x2. Then g'(x) = 4x3 – 12x and g”(x) = 12x2 – 12 = 12(x – 1)(x + 1). The sign is positive for x less than -1, negative between -1 and 1, and positive beyond 1. So both change. The inflection points are (-1, -5) and (1, -5).
What is a typical error?
Stopping at f” = 0. For y = x4, f” = 12x2 is zero at 0, but it never changes sign, so there is no inflection. Mathbot always runs the sign test.
Example: x^4 – 6x^2 has inflection points (-1, -5) and (1, -5)