We explore the geometric meaning of the third derivative by seeing what happens when it is positive/negative. Our geometric interpretation can be stated rigorously, and we provide a simplified proof of this.

Further reading:
Byerley, C., & Gordon, R. A. (2007). Measures of Aberrancy. Real Analysis Exchange, 32(1), 233266.
Gordon, R. A. (2004). The aberrancy of plane curves. The Mathematical Gazette, 89(516), 424436.
Schot, S. H. (1978). Aberrancy: Geometry of the third derivative. Mathematics Magazine, 51(5), 259275.

00:00 Intro
00:36 Intuitive sketches
02:24 Geometric interpretation
04:24 Simplifying assumptions
06:18 An equivalent geometric setup
08:01 Statement of result
08:52 Proof
11:20 Third derivative = 0 case