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**Note **To understand this section, you should be familiar with derivatives. Press the "All Tutorials" button on the sidebar to select one of the on-line tutorials on derivatives.

Let us start by looking at the definition in the textbook (and also in the Chapter Summary)

Maxima and Minima
f has a |

Sometimes it is not a simple matter to tell from the graph *Exactly* where the local extrema are situated. For instance, try graphing the curve y = x^{3}(x^{1/2} -1) for 0 x 2, and see if you can tell exactly where the absolute minimum lies. (This one is discussed in the on-line review exercises).

To help us locate extrema accurately, we classify them into three types and use calculus to assist us in locating them.

Locating Candidates for Relative Extrema
If f is continuous on its domain and differentiable except at a few isolated points, then its relative extrema occur among the following types of points.
The following figure shows several instances of all three types. |

Example 2 in the book goes applies this to the function f(x) = x^{4} - 4x^{3} we looked at above. Instead of repeating that example, let us go through a different example.

*Stationary Points* Set the derivative f'(x) = 0 and solve for x.

You should get three solutions (enter them in any order).

Now classify these three stationary points as relative maxima, minima, or neither. (Make all 6 selections and press "Check".)

*Singular points* To locate singular points, we look for values x where f'(x) is *not *defined, but f(x) *is *defined. However, f'(x) = 4x - 4x^{3} is defined for *every* x in the domain. Thus, there are no singular points.

*Endpoints* These are the endpoints of the domain, if any. Since the domain is [-1, +), it has only one end-point: x = -1.

**Q **Looking at the graph, we see that, at x = -1, f has a

**Q **OK what about those mysterious singular points? I haven't seen any yet.
**A **Try the following example.

Notice that something funny seems to be going on around x = 1. (We will zoom in later, after we have done the calculations.)

Try to identify the location of the statinoary point(s) on the graph before going further.

*Singular points* To locate singular points, we look for values x where f'(x) is *not *defined, but f(x) *is *defined. Now, you should laready have calculated f'(x) to answer the last question. It is

f'(x) = | 3(x-1) ^{1/3} | - 3 |

Notice that the denominator is zero when x = 1, so that f'(x) is not defined when x = 1, even though f(x) *is* defined when x = 1 (You should calculate f(1) = 0). Thus, we have a singular point at x = 1.

*Endpoints *The only endpoint is x = 0.

Here is a zoomed-in portion of the graph.

Notice that we have a relative minimum at the singular point x = 1, and a relative maximum at the stationary point a little to its right.

You now have several options

- Try some of the questions in the true/false quiz (warning: it covers the whole of Chapter 5) by pressing the button on the sidebar.
- Try some of the on-line review exercises (press the "Review Exercises" button on the sidebar. Questions 1 and 2 are relevant, but many of the others are based on Section 5.2.)
- Try some of the exercises on Section 5.1 of
*Applied Calculus,*or Section 13.1 of*Finite Mathematics and Applied Calculus*