SL 5.8 Maximum and Minimum
Content
- The second derivative.
- 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.