Turning points

There are three types of turning points:

Local maxima
Local minima
Points of inflection

We know that when f′(x) = 0 there will be a maximum or a minimum. Whether it is a maximum or minimum should be evident from looking at the graph of the original function. If a graph is not available, we can find out by plugging in a slightly smaller and slightly larger value than the point in question into f′(x). If the smaller value is negative and the larger value positive then it is a local minimum. If the smaller value is positive and the larger value negative then it is a local maximum.

If you take the derivative of a derivative function (one you have already derived) you get the second derivative. In mathematical notation, the second derivative is written as

We can use this to determine whether a point on a graph is a maximum, a minimum or a point of inflection as demonstrated in the following Figure 4.1.

Figure 4.1: Graph that shows a local maximum, a local minimum and points of inflection

Note: Notice how the points of inflection of f (x) are minima and maxima in f ′(x) and thus equal 0 in f ′′(x)

Finding turning points

The function f(x) = x³ + x² − 5x − 5 is shown. Use the first and second derivative to find its turning points: the minima, maxima and points of inflection (POI).