THE WEIERSTRASS THEOREM
Analysis
1. STATING THE THEOREM

The Weierstrass Theorem: If a function f(x) is defined and is continuous over a closed bounded interval [a, b], then f(X) has at least one absolute maximum and minimum point in the interval [a, b]. 

 

2. PLAYING WITH THE THEOREM. EXERCISES.

This first window is similar to the first page of the Bolzano's theorem, where a new control point called y has been added (a green circle with a red centre) which can move a green horizontal line. We know the height of this line which thus allows us to work out approximately where the maximum and minimum points are and the approximate values of these points.

All the graphs of functions in this window are differentiable and therefore are continuous along the complete real line and especially over any closed bounded interval. Therefore, they all satisfy the Weierstrass theorem hypotheses and consequently the thesis. In other words, all of them have absolute maximum and minimum points in any closed bounded interval.

 

1.- Given the function f(x)=(3x4-4x3-12x2)/40, is the Weierstrass theorem hypothesis satisfied in the interval [-1, 2.5]? If so, at which points in the interval does the function reach maximum and minimum values? What are these values approximately?

It is possible to have more than one maximum or minimum point, as the following example shows:

2.- The function f(x)=(x4-8x2)/40 satisfies the Weierstrass theorem hypothesis in the interval [-3.5, 3.5]. At which points in the interval are the maximum and minimum values located? (These points may occur at the endpoints of the interval). Find their approximate values.

Even if the hypothesis is not satisfied, i.e. if the graph is not continuous, the thesis may or may not be satisfied. Let's see an example in the second window on the previous page:

3.- Draw the graph of the function f(x)=1/x. Let p=0, q=1, r=0, s=1, t=0. Which Weierstrass theorem hypothesis is not satisfied in the interval [-3, 3]? Is the thesis satisfied?

There are two examples in the third window where the hypothesis is not satisfied but the thesis is.

The two graphs in the third window represent two functions: the first (in red) is the function of the sign of x i.e. equal to -1 when x<0, 0 when x is 0 and 1 when x>0. Look at it over the interval [-2 ,2]. In the second graph (in blue) g(x) is the square root of (x2-1); which you can see over the interval [-3, 3].

In this window we can study two different examples where the hypothesis of continuity is not satisfied.

 

As we are working with elementary functions, they only cease to be continuous at the points where they are not defined.

 

Let's continue with some exercises:

1.- Are the Weierstrass theorem hypotheses satisfied by the function f(x) over the interval [-2, 2]? What about the thesis? If so, where are the maximum and minimum points located? What are the values of these points?

2.- Now answer the questions above for the function g(x) over the interval [-3, 3].
       
           
  Valerio Chumillas Checa
 
Spanish Ministry of Education. Year 2001