Further Applications of Differentiation (DP IB Maths: AI HL)

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Paul

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Stationary Points & Turning Points

What is the difference between a stationary point and a turning point?

  • A stationary point is a point at which the gradient function is equal to zero
    • The tangent to the curve of the function is horizontal
  • A turning point is a type of stationary point, but in addition the function changes from increasing to decreasing, or vice versa
    • The curve ‘turns’ from ‘going upwards’ to ‘going downwards’ or vice versa
    • Turning points will either be (local) minimum or maximum points
  • A point of inflection could also be a stationary point but is not a turning point

How do I find stationary points and turning points?

  • For the function y equals f left parenthesis x right parenthesisstationary points can be found using the following process

STEP 1
Find the gradient function,space fraction numerator straight d y over denominator straight d x end fraction equals f apostrophe left parenthesis x right parenthesis

STEP 2
Solve the equationspace f apostrophe left parenthesis x right parenthesis equals 0 to find the x-coordiante(s) of any stationary points

STEP 3
If thespace y-coordinates of the stationary points are also required then substitute thespace x-coordinate(s) intospace f left parenthesis x right parenthesis

 

  • A GDC will solvespace f apostrophe left parenthesis x right parenthesis equals 0 and most will find the coordinates of turning points (minimum and maximum points) in graphing mode

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Testing for 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. space f apostrophe left parenthesis x right parenthesis equals 0
  • A local minimum point,space left parenthesis x comma space f left parenthesis x right parenthesis right parenthesis will be the lowest value ofspace f left parenthesis x right parenthesis in the local vicinity of the value ofspace x
    • The function may reach a lower value further afield
  • Similarly, a local maximum point, space left parenthesis x comma space f left parenthesis x right parenthesis right parenthesis will be the highest value ofspace f left parenthesis x right parenthesis in the local vicinity of the value ofspace x
    • The function may reach a greater value further afield
  • The graphs of many functions tend to infinity for large values ofspace x
    (and/or minus infinity for large negative values ofspace x)
  • The nature of a stationary point refers to whether it is a local minimum point, a local maximum point or a point of inflection
  • A global minimum point would represent the lowest value ofspace f left parenthesis x right parenthesis for all values ofspace x
    • similar for a global maximum point

How do I find local minimum & maximum points?

  • The nature of a stationary point can be determined using the first derivative but it is usually quicker and easier to use the second derivative
    • only in cases when the second derivative is zero is the first derivative method needed
  • For the functionspace f left parenthesis x right parenthesis
STEP 1

Findspace f apostrophe left parenthesis x right parenthesis and solvespace f apostrophe left parenthesis x right parenthesis equals 0 to find the x-coordinates of any stationary points

STEP 2 (Second derivative)

Findspace f apostrophe apostrophe left parenthesis x right parenthesis and evaluate it at each of the stationary points found in STEP 1

STEP 3 (Second derivative)
    • Ifspace f apostrophe apostrophe left parenthesis x right parenthesis equals 0 then the nature of the stationary point cannot be determined; use the first derivative method (STEP 4)
    • Ifspace f apostrophe apostrophe left parenthesis x right parenthesis greater than 0 then the curve of the graph ofspace y equals f left parenthesis x right parenthesis is concave up and the stationary point is a local minimum point
    • Ifspace f apostrophe apostrophe left parenthesis x right parenthesis less than 0 then the curve of the graph ofspace y equals f left parenthesis x right parenthesis is concave down and the stationary point is a local maximum point
STEP 4 (First derivative)

Find the sign of the first derivative just either side of the stationary point;  i.e. evaluatespace f apostrophe left parenthesis x minus h right parenthesis andspace f apostrophe left parenthesis x plus h right parenthesis for smallspace h

    • A local minimum point changes the function from decreasing to increasing
      • the gradient changes from negative to positive
      • space f apostrophe left parenthesis x minus h right parenthesis less than 0 comma space space f apostrophe left parenthesis x right parenthesis equals 0 comma space space f apostrophe left parenthesis x plus h right parenthesis greater than 0
    • A local maximum point changes the function from increasing to decreasing
      • the gradient changes from positive to negative
      • space f apostrophe left parenthesis x minus h right parenthesis greater than 0 comma space space f apostrophe left parenthesis x right parenthesis equals 0 comma space space f apostrophe left parenthesis x plus h right parenthesis less than 0

Stationary Points incr decr min max

  • A stationary point of inflection results from the function either increasing or decreasing on both sides of the stationary point
    • the gradient does not change sign
    • space f apostrophe left parenthesis x minus h right parenthesis greater than 0 comma space space f apostrophe left parenthesis x plus h right parenthesis greater than 0   or  space f apostrophe left parenthesis x minus h right parenthesis less than 0 comma space space f apostrophe left parenthesis x plus h right parenthesis less than 0
    • a point of inflection does not necessarily havespace f apostrophe left parenthesis x right parenthesis equals 0
      • this method will only find those that do - and are often called horizontal points of inflection

Stationary Points point of inflection

Examiner Tip

  • Exam questions may use the phrase “classify turning points” instead of “find the nature of turning points”
  • Using your GDC to sketch the curve is a valid test for the nature of a stationary point in an exam unless the question says "show that..." or asks for an algebraic method
  • Even if required to show a full algebraic solution you can still use your GDC to tell you what you’re aiming for and to check your work

Worked example

Find the coordinates and the nature of any stationary points on the graph ofspace y equals f left parenthesis x right parenthesis wherespace f left parenthesis x right parenthesis equals 2 x cubed minus 3 x squared minus 36 x plus 25.

5-2-4-ib-sl-aa-only-we-soltn

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.