SL 5.7—Second derivative and graph behaviour
- Syllabus
- First assessment 2021
- Objective
- —
- Level
- HL
The second derivative describes curvature and acceleration
The second derivative describes curvature and acceleration.
f″ measures how the slope changes; its sign indicates concavity and, in kinematics, acceleration.
Example
For f(x)=x³, f″=6x, so the graph is concave down for x<0 and up for x>0.
Use f′ for increasing/decreasing and f″ for curvature; test continuity when claiming an inflection.
A positive second derivative does not mean the function itself is positive.