For example plot f ( x) = x 3 and note that f ′ is zero at x = 0, yet it is neither a relative maximum nor a relative minimum.

Relative extrema on a graph. To find the exact values (coordinates) for these points, use the three. Now let's do an example dealing with absolute extrema. The word “extrema” is plural for the word “extremum”.

The other type of extrema are the local (or relative) extrema. The relative extrema of the function f(x) = 5x 3 + x 2 shown as colored dots on its graph. The x guy is where the max occurs.

Look back at the graph. F has a relative max of 1 at x = 2. Officially, for this graph, we'd say:

The relative maximum and minimum values comprise the relative extrema of \(f\). Relative extrema is a fancy term for the maximum and minimum points of a graph, relative to nearby points. So we start with differentiating :

Notice that f ′ ( x) does not change. Here the relative extrema are in red with x 2 a local max and x 3 a local min. We still do not have the tools.

The value of x within the domain of f (x), which is neither a local maximum nor a local minimum, is called the point of. In higher dimensions, saddle points are another example of critical. These are points that represent extreme values in some small ’neighborhood’ of the function.