Introduction

Relevance of the Topic

"Focusing on Maximum and Minimum of the Second-Degree Function is the key to unlocking advanced understanding of many mathematical concepts. The reason behind this is threefold. First, it returns to the heart of mathematics - the relationship between variables. Second, it deals with optimization concepts - an area of extreme practical importance in physical sciences, economics, engineering, among other fields. Third, it delimits the field of the parabola - one of the most fundamental and iconic forms in mathematics."

Contextualization

"The Second-Degree Function holds a special place in the High School Mathematics curriculum. Being one of the first nonlinear functions students encounter, it paves the way for the study of other polynomial functions. Knowledge of second-degree functions and their properties is also vital for the study of differential and integral calculus. Mastering the calculation of Maxima and Minima in this function allows for a graphical and analytical interpretation of the behavior of phenomena modeled by it."

Theoretical Development

Components

• Second-Degree Function: It is emphasized in this topic that a second-degree function is any function that can be expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants, with a ≠ 0. The important point here is that the parabola generated by this function, when a > 0, has concavity facing upwards and, when a < 0, has concavity facing downwards.

• Maxima and Minima of the Second-Degree Function: The maximum and minimum points, that is, the vertices of the parabolas, are the critical points of this type of function. They are where the function reaches its maximum or minimum value. They represent one of the main focuses of the study of the second-degree function and are calculated by the expression -b/2a.

• Problem Solving: After understanding the concept and identifying the maximum and minimum points, the application for solving practical problems is worked on. This is a crucial point because it is where theory meets direct application.

Key Terms

• Parabola: is the curve that graphically represents the second-degree function.

• Vertex of the Parabola: is the point of maximum (for parabolas with the opening downwards) or minimum (for parabolas with the opening upwards) of the second-degree function. It is the point where the parabola changes direction.

• Critical Point: It is the vertex of the parabola, that is, the point of the function where the maximum or minimum value occurs.

Examples and Cases

• Example 1 - Increasing Function: Consider the function f(x) = x² - 4x + 3. In this function, we have a = 1, b = -4, and c = 3. The vertex of this parabola is calculated by the expression -b/2a, which gives us 2. This is the value of x that results in the function's minimum.

• Example 2 - Decreasing Function: Consider the function f(x) = -x² + 4x - 1. In this function, we have a = -1, b = 4, and c = -1. The vertex of this parabola can be calculated by the expression -b/2a, also resulting in 2. However, unlike the previous example, this is the value of x that results in the function's maximum.

• Example 3 - Practical Problem: A rectangle has a perimeter of 36 meters. What should be its dimensions for the area to be maximum? Using the expression for the area of a rectangle, A = L * W, and the expression for the perimeter, P = 2(L + W) - which can be rearranged to obtain W = P/2 - L - we can substitute the width in the area formula to obtain the area function in terms of the length: A(L) = L(P/2 - L). Then, we can calculate the value of L for which the area is maximum using the expression -b/2a.

Detailed Summary

Key Points

• Definition and Components: The Second-Degree Function is any function expressed in the form f(x) = ax² + bx + c, a ≠ 0. It is represented by a parabola whose concavity depends on the sign of a.

• Maxima and Minima Points: They are the vertices of the parabola, representing where the function reaches its maximum or minimum value. The maximum or minimum points are calculated by the expression -b/2a.

• Application Context: The second-degree function is fundamental in the study of other polynomial functions and in concepts of differential and integral calculus. The calculation of maxima and minima introduces optimization concepts.

Conclusions

• Properties of the Second-Degree Function: Its graphical representation, the parabola, is determined by the coefficients; the concavity is given by the coefficient a; maximum or minimum is given by the vertex of the parabola.

• Importance of Study: Understanding the second-degree function well is essential for progression in mathematics studies, particularly in more advanced topics such as calculus and optimization.

Exercises

1. Exercise 1: Determine the values of x that result in the maximum and minimum points of the function f(x) = 2x² - 8x + 7.

2. Exercise 2: Given that f(x) = -x² + 10x - 21, find the critical point and determine if it represents a maximum or minimum of the function.

3. Exercise 3: A company wants to build a rectangular warehouse on an available land of 300 meters in length. What should be the width of the warehouse for its area to be the largest possible?