Here \(f'\) vanishes at \(x=\pm1\). Its signs are \(+, -, +\), so \(f\) has a maximum at \(x=-1\) and a minimum at \(x=1\). Since \(f''(x)=6x\), the change of concavity occurs at \(x=0\).

A derivative graph does not determine the vertical position of \(f\). An additional point or initial value is needed. Do not confuse the height of \(f'\) with its gradient.

Concavity of Graphs

Applications of Differentiation

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Reading Derivative Graphs

Reading Derivative Graphs