Unit 3

Curve Sketching

Lesson 3: Maximum and Minimum Values

In the previous Lesson, the focus was on a function’s intervals of increase and decrease. In other words, the focus was on the behaviour of the function on either side of the critical points, where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» or where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»D«/mi»«mi»N«/mi»«mi»E«/mi»«/mrow»«/mstyle»«/math». In this Lesson, the focus will be on the behaviour of the function at those critical points.

Interactive

Interactive

---

Click the interactive button to open the applet Where is a Derivative Zero? The applet shows a polynomial function that can be transformed by moving the points on the graph.

• Predict where the derivative of this function will be zero.
• Use the checkboxes to check your prediction.
• Move some points on the graph of the original function and try again.

What characteristic(s) on the graph of the original function allow you to determine where the derivative of that function will be zero?