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Further Applications of Differentiation (DP IB Maths: AI HL)
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
- A GDC will solve 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.
- 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 …
Find and solve to find the -coordinates of any stationary points
Find and evaluate it at each of the stationary points found in STEP 1
-
- 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
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 local minimum point changes the function from decreasing to increasing
- 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
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 of where.
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