On the following page, we will look at graphs and derivatives. We will get some practice in sketching gradient functions and we will carefully consider stationary points (maximum, minimum and points of inflexion) as well as non-stationary points of inflexion.
Key Concepts
On this page, you should learn about
graphs of \(f,f'\ \mathrm{and} \ f''\)
local maximum and minimum points
stationary points of inflexion
non-stationary points of inflexion
Essentials
The following videos will help you understand all the concepts from this page
Sketching the Gradient Function
In the following video we shall see how to sketch graphs of a gradient function
You can get some practice of sketching gradient functions by trying the activity below. This works best if you have a device with a tactile screen so that you can sketch the graphs on screen. If this is not the case, you can print a copy from here and draw with a pencil.
Choose the pen tool and sketch the gradient function
Check your answer by clicking in the check box
Turn off the answer and delete your sketch before moving on to the next question
Stationary Points
In the following section we consider stationary points, that is, points on a graph when the gradient = 0.
There are three different types of stationary points
Local Maxima
Local Minima
Point of Inflexion
Local Maxima and Local Minima
Drag the point and explore the gradient of this function
Stationary Point of Inflexion
Drag the point and explore the gradient of this function
In the following video, we shall summarize the main features of graphs with stationary points and how we use the gradient function (and the gradient of the gradient function) to determine the nature of the stationary points.
Stationary Points Example
In the following video, we will look at an example in which we are required to find the coordinates of stationary points
Find the co-ordinates of the stationary points on the curve \(y = x^4 – 4x^3\) and determine their nature. Sketch the curve.
Non-stationary points of inflexion exist where \(\frac{d^2y}{dx^2}=0\) and \(\frac { dy }{ dx } \neq 0\)
In the following video, we see why this is the case and we look at the following example in which we are required to find the coordinates of a point of inflexion.
Find the coordinates of the point of inflexion on the curve \(y=x^3-6x^2+13x-9\)