How to locate stationary points of a function.
For a function: y = f(x), a stationary point is a point on the function graph where the gradient of the function is zero.
If the gradient of the function changes sign at the stationary point, then it is called a turning point, which can be a local maximum or local minimum:
If the gradient of the function does not change sign at the stationary point, then it is a point of horizontal inflection:
The derivative of the function y = f(x) gives us the gradient of any point, so that if we know the derivative, dy/dx, then we can locate the stationary points of the function y = f(x) without even drawing a graph. Also, by evaluating dy/dx just before and just after each stationary point, we can find out if it is a local maximum, local minimum, or point of horizontal inflection.
NOTE: The second derivative of f(x): d2y/dx2, is often used to verify that a point is a local maximum or local minimum. The sign of the second derivative at the stationary point is positive for a local minimum, and negative for a local maximum.
Maths Helper Plus can greatly simplify the process of finding stationary points on function graphs. Here are some of the ways it can help:
In the 'Method' section below, we will use Maths Helper Plus to locate the stationary points of the function:
Step 1 Start with an empty Maths Helper Plus document
If you have just launched the software then you already have an empty document, otherwise hold down ‘Ctrl’ while you briefly press the ‘N’ key.
Step 2 Graph the function
For more help on entering functions, see the 'Easy Start' tutorial in the Maths Helper Plus 'help'. To access the tutorial, click 'Help' on the Maths Helper Plus menu bar, then select: 'Tutorial...'
If the required area between the function graphs is not completely
visible, you need to adjust the
For more help on setting graph scales, click 'Help' on the Maths Helper Plus menu bar, then select 'Index'. Click the button at the top left corner of the help screen. Click 'Training', then 'Essential skills', then 'Graph scale magic'.
Step 3 Plot the first derivative graph
Carefully point to one of the function curves with the mouse pointer. Double click to display the options dialog box for that function. (This may work better on parts of the graph that are not so steep.)
The 'Plot settings' tab will be displayed. See below:
Click to select the dy/dx option. The first derivative graph will display in red on the graph view.
Click OK to close the options dialog box.
Step 4 Locate the stationary points
Type 1 - Local maximum or local minimum
The 'turning point tool' in Maths Helper Plus will locate a local maximum or local minimum in most cases.
This diagram illustrates the procedure:
The steps are:
For the example y = 0.1x³ - 0.2x² - 1.5x + 1, stationary points are located at these points:
For the point: (-1.66667,2.48148), we observe from the dy/dx graph that dy/dx is positive to the left of the point, and negative to the right of the point. This is consistent with a local maximum.
For the point (3,-2.6), dy/dx is negative to the left of the point and positive to the right of the point, which is consistent with a local minimum.
Type 2 - Point of horizontal inflection
More about points of horizontal inflection:
A point of inflection on y = f(x) is a point where the gradient stops increasing and starts decreasing, or stops decreasing and starts increasing.
Points of inflection can be recognised from the dy/dx graph. The dy/dx graph has a local maximum or local minimum with the same 'x' value as a point of inflection on the y = f(x) graph.
The 'turning point tool' in Maths Helper Plus will not locate a point of inflection on the graph of y = f(x). We locate it indirectly by first locating the corresponding local maximum or minimum on the graph of dy/dx. If the 'y' coordinate of this point is zero, then the 'x' coordinate is the same as the 'x' coordinate of a point of horizontal inflection on the graph of y = f(x)
For the example graph, the following local minimum was located on the dy/dx graph:
This means that the function y = 0.1x³ - 0.2x² - 1.5x + 1 has an inflection point at x = 0.666666, but it is not a point of horizontal inflection, because the 'y' coordinate is not zero.