Stationary Points (DP IB Analysis & Approaches (AA)): Revision Note

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 , stationary points can be found using the following process

STEP 1

Find the gradient function,

STEP 2

Solve the equation to find the -coordiante(s) of any stationary points

STEP 3

If the-coordinates of the stationary points are also required then substitute the-coordinate(s) into

• A GDC will solve and most will find the coordinates of turning points (minimum and maximum points) in graphing mode

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. 

• 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 highest 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 point, a local maximum point or a point of inflection

• A global minimum point would represent the lowest value of for all values of

  • 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 function …

STEP 1

Find and solve to find the -coordinates of any stationary points

STEP 2 (Second derivative)

Find and evaluate it at each of the stationary points found in STEP 1

STEP 3 (Second derivative)

• If then the nature of the stationary point cannot be determined; use the first derivative method (STEP 4)

• If then the curve of the graph of is concave up and the stationary point is a local minimum point

• If then the curve of the graph of 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. evaluate and for small

• A local minimum point changes the function from decreasing to increasing

  • the gradient changes from negative to positive

  • 

• A local maximum point changes the function from increasing to decreasing

  • the gradient changes from positive to negative

  • 

• 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

  •    or  

  • a point of inflection does not necessarily have

    • this method will only find those that do - and are often called horizontal points of inflection