Stationary Points and Second Derivatives

Aims: To use the second derivative to decide on the “nature” of a stationary point.

To look at questions on optimisation and why the stationary points are important.

Lesson Outcomes Name: What is a second derivative?Describe: How to find the second derivative of

a function.Explain: How you can use the second

derivative of a function along with the x-coordinate of one of its stationary points to decide if the point is a maximum or minimum point.

Do questions on optimisation.

Stationary PointsStationary points are the points when the gradient

is 0. The points where the function is not increasing or decreasing (hence the term stationary).

Stationary Points fall into one of three types.

Determining The NatureWhile we may be able to know the nature for

simple graphs there is a numerical method that can help us.

We find the “Second Derivative” which is the gradient of the gradient and tells how the gradient is changing at that point.

We write...

)(''or 2

2

xfdx

yd

Determining The NatureThinking about the

gradient again.When the graph is

maximum its gradient is zero but is it about to increase or decrease?

When the graph is minimum its gradient is zero but is it about to increase or decrease?

Determining The NatureThis tells us that a

maximum point will have a second derivative that is decreasing (negative) and a minimum point will have a second derivative that is increasing (positive)

If the second derivative is 0 you need to investigate. (Substitute either side of the point.)

ExampleFind the stationary points of the curve y=f(x)

and determine their nature given that...

193)( 23 xxxxfDerivative Second Derivative

=0 Second Derivative at first x

Factorised Second Derivative at second x

x values Max at Min at

ActivitySort and Stick cards into groups completing

several versions of the problem just demonstrtated.