Expert reviewed • 05 March 2025 • 7 minute read
As noted in the previous chapter stationary points are points on the graph of a function where the derivative is zero. These points are significant because they can indicate where a function reaches a maximum or minimum, or where a change in the slope or direction of the graph occurs.
Turning points on a graph, are stationary points, which curve smoothly, changing the function from decreasing to increasing, or increasing to decreasing. The different types of stationary points are as follows:
As touched on in the previous chapter, to find any stationary points of a function, you must equate the function's derivative with zero. By doing this, you will be able to find the turning points, points of inflection, discontinuities and any other stationary points
Once this step has been completed, it is important to construct a table of signs, to determine the gradient of the slope. This is important when it comes to graphing, as you are able to determine which stationary points are the local maximum and minimum, while making it easier to understand the overall shape of the graph. If you are not graphing, a table of values is not always necessary.
Determine the stationary points of the following function:
Since we have solved for the derivative, we can equate it with zero and factorise, to determine the stationary points of the function.
Now that we have the x-values of the stationary points, we must substitute them back into the original equation to find the y-values.
The stationary points of the function are and