Stationary Points & Turning Points (Edexcel IGCSE Further Pure Maths)

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 derivative of a function is equal to zero

  • The tangent to the curve of the function is horizontal (gradient = 0)

• A turning point is a point at which

  • the derivative of a function is equal to zero

  • AND the derivative changes sign (from negative to positive, or positive to negative)

    • i.e. the curve changes from ‘going upwards’ to ‘going downwards’ (or vice versa)

  • Turning points will either be (local) minimum or (local) maximum points

• All turning points are also stationary points

  • But not all stationary points are turning points

How do I find stationary points and turning points?

• For the function , stationary points can be found using the following process
  

• STEP 1
  Find the derivative,
  

• STEP 2
  Solve the equation  

  • The solution(s) are the -coordinate(s) of any stationary points

    • Remember, 'stationary points' includes turning points
      

• STEP 3
  Find the corresponding -coordinates (if necessary)

  • Substitute each-coordinate(s) into

  • If the question only asks for the -coordinates, you can skip this step!
    

• More work is needed to find if a stationary point is a turning point

  • i.e. if it is a (local) maximum or (local) minimum

  • See the following note

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 derivative at stationary points equals zero

    • i.e. 

  • But not all points with a zero derivative are maximum or minimum points

• A local minimum point, 

  • will have the lowest value of  in the local vicinity of the value

  • But may reach a lower value further away

• A local maximum point,  

  • will have the highest value of  in the local vicinity of the value

  • But may reach a higher value further away

• A global maximum (or global minimum) point

  • has the highest (or lowest) value of for all values of in the domain of 

    • A global maximum (or minimum) may not be a local maximum (or minimum)

    • and vice versa

  • Not all functions have a global maximum (or minimum)

    • e.g. some functions tend to infinity as  tends to infinity

    • or as  approaches some other value (vertical asymptotes)

How can I use the derivative to identify local minimum & maximum points?

• The nature of a stationary point can be determined using the derivative
  

• For the function …
  

• STEP 1
  Find and solve 

  • The solutions are the the -coordinates of any stationary points

• STEP 2
  Find the sign of the derivative just either side of each stationary point

  • i.e. evaluate and  for small

    • Choose a convenient value for 

    • Just make sure you don't 'jump over' any other stationary points!

• At a local minimum point

  • the derivative changes from negative to positive

  • 

• At a local maximum point 

  • the derivative changes from positive to negative

  • 

    

• If the derivative does not change sign at the point

  • (i.e. if it is positive on both sides or negative on both sides)

  • then the point is not a local minimum or a local maximum

  • But it is still a stationary point if  there

What is the second derivative?

• It is possible to differentiate a function more than once

  • If you differentiate , you find its (first) derivative  

    • e.g. the (first) derivative of is

  • Then you can differentiate the derivative again to find the second derivative  

    • e.g. the derivative of  is 

    • so the second derivative of  is 

• The second derivative can be used to test the nature of a stationary point
  

How can I use the second derivative to identify local minimum & maximum points?

• STEP 1
  Find and solve 

  • The solutions are the the -coordinates of any stationary points

• STEP 2
  Find the second derivative

  • Do this by differentiating  from Step 1

• STEP 3
  Find the value of  at each of the stationary points

  • i.e., by substituting the -coordinate of each point into   

• If  
  

  • then the stationary point is a local minimum

• If   

  • then the stationary point is a local maximum

• If  

  • then the test does not tell you anything

  • the stationary point may be a local maximum, a local minimum, or neither

  • In this case, use the first derivative test instead