Abstract

This research is focusing on the extremum of a given function which involves triangle functions. We collect some typical examples from ordinary high school mathematics teaching and demonstrate several typical mistakes which were usually posed by ordinary high school students usually and analyze the reasons for the errors. These results will be used to help ordinary high school students to well understand AM-GM inequalities and related applications for maximum and minimum values of triangle type functions.

1. Introduction

The starting point of this investigation is to clarify some various errors in computing the maximum or minimum values of a given function which involves triangle types functions by using the arithmetic mean and geometry mean (AM-GM) inequalities ^{1, 2} which usually occur in high school mathematics study. Inequalities are one of the key tools to solve mathematical problems in applied sciences, and also play an important role in ordinary high school mathematics teaching and examinations. So we collect three typical related problems from high school mathematics teaching which aim to illustrate some typical mistakes happening usually by ordinary high school students and give some useful suggestions.

AM-GM inequalities read:

Let , and denote , . Then . Equalities occur if and only if .

Maximum value or minimum value can be obtained upon the fixed value of by using or by using respectively. The counterexample shows that the condition is necessary. Another counterexample, , although anyway, is wrong, so is necessary.

The paper is organized as follows: in the second chapter we present three typical examples and give error analysis, in the third chapter, we give a further analysis of the triangle type functions concerning maximum and minimum values, and give some useful suggestions for ordinary high school students.

2. Examples

The following examples were selected from problems or examinations in ordinary high school mathematics, which vividly describe the typical error in the applications of AM-GM inequalities.

For any ,,, the inequality

(1)

is always holds, then the minimum value of the entire number is ( ):

A. 1. B. 2. C. 3. D. 4

The error answer. Because , is an odd function, hence we rewrite the inequality

as

Noting that is monotone increasing on R, we know so Furthermore, by the knowledge of trigonometric functions, we obtain the following,

(2)

where But therefore,

So , then the minimum value of is .

Remark 1. Here we have a key question whether the function can equal 1 or not. The error is that the conditions and cannot be held at the same time. Now we give the right answer as follows.

The right answer. Noting that is an odd function, the inequality (1) becomes

Sinceis monotone increasing, we have then

Defining , we aim to compute the maximum value of . Differentiating , we have

Let we have Noting that if so is monotone increasing; and if ,, hence is monotone decreasing. Thus, the maximum value of is then the minimum value of equals 1.

Remark 2. The right answer used the monotonicity of the function to transform the inequality and used the derivative function to obtain the maximum value, thus obtaining the value of .

Remark 3. When in (2), then But if then

so . Then

Therefore, in this situation, we have

Remark 4. According to ,, the inequality is transformed into the following form:

(3)

But the solving of inequality (3) is very complicated, and this is not recommended, so we omit it here.

Suppose that , compute the maximum value of .

The error answer. By the AM-GM inequality, we have

Thus when or , .

Remark 5. The error answer is wrong because if , or , then , but it contradicts with .

The right answer. Because

then Thus we have or Therefore, when where we have when we have therefore,

We shall give another right answer as follows.

By the AM-GM inequality, we have

Hence thus where

Compute the maximum value of the function

The wrong answer.

(4)

Remark 6. In the wrong answer above, the necessary condition of AM-GM inequalities has been missed. Here should be it follows that Therefore the maximum value of but it is wrong.

The right answer. By AM-GM inequalities, we have the following.

(5)

Remark 7. In (4), while applying the AM-GM inequalities, we should convert the product of to a fixed constant number, such as

otherwise yields the wrong results, for instance, see (4), which is not a constant number. Another solution is as follows:

3. Conclusions

We discuss three typical examples aiming to find the extremum of a giving function that involves triangle-type functions in high school mathematics teaching and give some useful suggestions for the high school students who are hard to understand the problems of AM-GM inequalities and their applications. Students should pay much attention to the domain of triangle type functions in the problems and the necessary and sufficient conditions of AM-GM inequalities. Given a function on a given interval, to find the maximum or minimum, the general method is to use the algebraic and geometric properties of the function, or use derivatives to obtain the monotonicity of the function on the intervals. Furthermore, it should be noted that if the function is composed of two functions, e. x.,

But the extremum of function is not necessarily equal to the sum, the difference, the product, the quotient, or the composition of the maximum or minimum values of the function and .

References