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

Local minimum & maximum points

What are local minimum and maximum points?

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

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

    • I.e. $f^{\text{'}}\left(x\right)=0$

• A local minimum point, $(x,f(x\left)\right)$will be the lowest value of$f\left(x\right)$ in the local vicinity of the value of $x$

  • The function may reach a lower value somewhere further away

• A local maximum point, $(x,f(x\left)\right)$will be the greatest value of$f\left(x\right)$ in the local vicinity of the value of $x$

  • The function may reach a greater value somewhere further away

• 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 $y=f\left(x\right)$
   

• STEP 1
  Plot the graph of $y=f\left(x\right)$
  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 $x$ and $y$ 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

  • At a local minimum the function changes from decreasing to increasing

    • the gradient changes from negative to positive

  • At a local maximum the function changes from increasing to decreasing

    • the gradient changes from positive to negative