5.1 Maxima and Minima

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Note To understand this section, you should be familiar with derivatives.

Let us start by looking at the definition.

Maxima and Minima

f has a relative maximum at c if there is some interval (r, s) (even a very small one) containing c for which f(c) f(x) for all x between r and s for which f(x) is defined.


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f has a relative minimum at c if there is an interval (r, s) (even a very small one) containing c for which f(c) f(x) for all x between r and s for which f(x) is defined.

By a relative exremum, we mean either a relative maximum or a relative minimum.

The following figure shows several relative extrema.

Here is the graph of f(x) = 3x4 - 4x3 with domain [-1, +) (discussed in Example 2 in the text).

Q f has:

    at (-1, 7)
    at (0, 0)
    at (1, -1)
    at (1.5, 1.6875)

Absolute Maxima and Minima

Relative extrema may sometimes also be absolute extrema, as the following definition shows.

f has an absolute maximum at c if f(c) f(x) for every x in the domain of f.
f has an absolute minimum at c if f(c) f(x) for every x in the domain of f.

The following figure shows several relative extrema that are also absolute extrema.

Note
All absolute extrema are automatically relative extrema, according to our convention.

Look again at the graph of f(x) = 3x4 - 4x3 with domain [-1, +)

Q The function f has

    extremum at (-1, 7)
    extremum at (0, 0)
    extremum at (1, -1)
    extremum at (1.5, 1.6875)

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 = x3(x1/2 -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.

1. Stationary points: points x in the domain where f'(x) = 0. To locate stationary points, set f'(x) = 0 and solve for x.
2. Singular points: points x in the domain where f'(x) is not defined. To locate singular points, find values x where f'(x) is not defined, but f(x) is defined.
3. Endpoints: the endpoints of the domain, if any. Recall that closed intervals contain endpoints, but open intervals do not.

The following figure shows several instances of all three types.

Example 2 in the book goes applies this to the function f(x) = x4 - 4x3 we looked at above. Instead of repeating that example, let us go through a different example.

Here is the graph of f(x) = 2x2 - x4, with domain [-1, +).

Stationary Points Set the derivative f'(x) = 0 and solve for x.
You should get three solutions (enter them in any order).

    x = ,   ,      

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

f has at the point where x =
f has at the point where x =
f has at the point where x =
 

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 - 4x3 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

    Minimum Maximum Both
    Neither Impossible to say

Q Finally,

    f has
    and a

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

Here is the graph of f(x) = (x-1)2/3 - 3(x-1), with domain [0, +).

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

Stationary Points (Set the derivative f'(x) = 0 and solve for x.)
Enter as many different stationary points as you calculate there are, from left to right, leaving the rest blank (no repeats, please). Use a fraction or decimals. Decimal approximations should be accurate to at least four digits.

x = ,   ,      

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) =
    2

    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.

Acknowledgement: The above is based on class materials from Stefan Waner and Steven R. Costenoble , for more content of the author, please visit here.