5.2.4 Further Applications of Differentiation

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 exhibits this property, 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

Find the gradient function,

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

If the-coordaintes 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

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 lowest 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 the nature of a stationary point?

• 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 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