Local Minimum & Maximum Points (DP IB Applications & Interpretation (AI)): Revision Note

Local Minimum & Maximum Points

What are local minimum and maximum points?

• Local minimum and maximum points are two types of stationary point

  • The gradient function (derivative) at such points equals zero
    i.e.

• A local minimum point, will be the lowest value of in the local vicinity of the value of 

  • The function may reach a lower value further afield

• Similarly, a local maximum point, will be the greatest value of in the local vicinity of the value of 

  • The function may reach a greater value further afield

• The graphs of many functions tend to infinity for large values of 
  (and/or minus infinity for large negative values of )

• The nature of a stationary point refers to whether it is a local minimum or local maximum point

How do I find the coordinates and nature of stationary points?

• The instructions below describe how to find local minimum and maximum points using a GDC on the graph of the function .

 STEP 1

 Plot the graph of 
  Sketch the graph as part of the solution

 STEP 2

 Use the options from the graphing screen to “solve for minimum”

 The GDC will display the  and  coordinates of the first minimum point

 Scroll onwards to see there are anymore minimum points

 Note down the coordinates and the type of stationary point

 STEP 3

 Repeat STEP 2 but use “solve for maximum” on your GDC

• In STEP 2 the nature of the stationary point should be easy to tell from the graph

  • a local minimum changes the function from decreasing to increasing

    • the gradient changes from negative to positive

  • a local maximum changes the function from increasing to decreasing

    • the gradient changes from positive to negative