Graphical behaviour of functions, including the relationship between the graphs of f,f′ and f″.
Points of inflexion with zero and non-zero gradients.
Understandings:
Concavity
points of inflection or inflexion points
Curve is rising.
Curve is falling.
Curve is concave up.
Curve is concave down.
second derivative test
Guidance, clarification and syllabus links
Use of both forms of notation, \(\frac{d^2y}{dx^2}\) and f″(x).
Technology can be used to explore graphs and calculate the derivatives of functions.
At a point of inflexion, f″(x)=0 and changes sign (concavity change), for example f″(x)=0 is not a sufficient condition for a point of inflexion for y=x4 at (0,0).
Use of the terms “concave-up” for f″(x)>0, and “concave-down” for f″(x)<0.