“Teach A Level Maths” Vol. 1: AS Core Modules 13: Stationary Points © Christine Crisp

Module C1 Module C2 AQA Edexcel OCR MEI/OCR "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

The stationary points of a curve are the points where the gradient is zero A local maximum x x A local minimum The word local is usually omitted and the points called maximum and minimum points.

e.g.1 Find the coordinates of the stationary points on the curve Solution: Tip: Watch out for common factors when finding stationary points. or The stationary points are (3, -27) and ( -1, 5)

Exercises Find the coordinates of the stationary points of the following functions 1. 2. Solutions: 1. Ans: St. pt. is ( 2, 1)

2. Solution: Ans: St. pts. are ( 1, -6) and ( -2, 21 )

We need to be able to determine the nature of a stationary point ( whether it is a max or a min ). There are several ways of doing this. e.g. On the left of a maximum, the gradient is positive On the right of a maximum, the gradient is negative

So, for a max the gradients are At the max On the left of the max On the right of the max The opposite is true for a minimum Calculating the gradients on the left and right of a stationary point tells us whether the point is a max or a min.

e. g. 2 Find the coordinates of the stationary point of the curve e.g.2 Find the coordinates of the stationary point of the curve . Is the point a max or min? Solution: Substitute in (1): On the left of x = 2 e.g. at x = 1, On the right of x = 2 e.g. at x = 3, We have is a min

Another method for determining the nature of a stationary point. e.g.3 Consider The gradient function is given by At the max of the gradient is 0 but the gradient of the gradient is negative.

Another method for determining the nature of a stationary point. e.g.3 Consider The gradient function is given by At the min of the gradient of the gradient is positive. The notation for the gradient of the gradient is “d 2 y by d x squared”

e.g.3 ( continued ) Find the stationary points on the curve and distinguish between the max and the min. Solution: Stationary points: is called the 2nd derivative or We now need to find the y-coordinates of the st. pts.

To distinguish between max and min we use the 2nd derivative, at the stationary points. max at At , min at

SUMMARY To find stationary points, solve the equation Determine the nature of the stationary points either by finding the gradients on the left and right of the stationary points maximum minimum or by finding the value of the 2nd derivative at the stationary points

Exercises Find the coordinates of the stationary points of the following functions, determine the nature of each and sketch the functions. 1. is a min. is a max. Ans. 2. is a min. is a max. Ans.

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

The stationary points of a curve are the points where the gradient is zero A local maximum A local minimum x The word local is usually omitted and the points called maximum and minimum points. e.g.

e.g.1 Find the coordinates of the stationary points on the curve Solution: or The stationary points are (3, -27) and ( -1, 5)

For a max we have The opposite is true for a minimum At the max On the right of the max On the left of the max Calculating the gradients on the left and right of a stationary point tells us whether the point is a max or a min. Determining the nature of a Stationary Point

At the max of the gradient is 0, but the gradient of the gradient is negative. The gradient function is given by e.g. Consider Another method for determining the nature of a stationary point.

The notation for the gradient of the gradient is “d 2 y by d x squared” At the min of The gradient function is given by the gradient of the gradient is positive.

The gradient of the gradient is called the 2nd derivative and is written as

e.g. Find the stationary points on the curve and distinguish between the max and the min. Solution: Stationary points: or We now need to find the y-coordinates of the st. pts.

max at min at At , To distinguish between max and min we use the 2nd derivative,

SUMMARY To find stationary points, solve the equation maximum minimum Determine the nature of the stationary points either by finding the gradients on the left and right of the stationary points or by finding the value of the 2nd derivative at the stationary points