Minimize F(x) = √(x² - 2x + 2) + √(x² - 4x + 29) - Maxima And Minima Problem Solved
Introduction
In this article, we delve into the fascinating realm of minima and maxima problems, specifically focusing on finding the least value of the function f(x) = √(x² - 2x + 2) + √(x² - 4x + 29). This problem elegantly combines the concepts of quadratic expressions and radicals, requiring a strategic approach to unravel its solution. Our discussion will encompass a detailed step-by-step analysis, employing both algebraic manipulations and geometric interpretations to arrive at the minimum value. Furthermore, we will pinpoint the exact value of x (denoted as α) where this minimum occurs. Finally, we will compute the value of [m] + [α], where m represents the least value of f(x) and [.] signifies the greatest integer function. This exploration is not just about finding the answer; it's about understanding the underlying principles and techniques applicable to a broader range of optimization problems. Understanding the interplay between algebraic expressions and their geometric counterparts is crucial for mastering such problems. This article aims to provide a comprehensive guide, suitable for students and enthusiasts alike, seeking to deepen their grasp of calculus and analytical problem-solving.
Problem Statement and Initial Approach
Let's restate the core problem: We are given the function f(x) = √(x² - 2x + 2) + √(x² - 4x + 29) and our objective is to determine its minimum value, which we'll denote as m. We must also find the value of x (α) at which this minimum occurs. The ultimate goal is to calculate [m] + [α], where [.] denotes the greatest integer function. The initial approach to tackling this problem involves recognizing the expressions within the square roots as potentially related to the equations of circles or distances. By completing the squares inside the radicals, we can rewrite the function in a more revealing form. This algebraic manipulation is a key step towards uncovering the geometric interpretation of the problem. This method allows us to visualize the function as the sum of distances, paving the way for a geometric solution. We can rewrite the expression inside the first square root as (x - 1)² + 1 and the expression inside the second square root as (x - 2)² + 25. These forms immediately suggest a connection to the distance formula in a two-dimensional coordinate system. This strategic transformation of the algebraic expression into a geometric representation is a common and powerful technique in problem-solving, and it's one we'll leverage throughout our analysis.
Algebraic Manipulation and Geometric Interpretation
To effectively solve this problem, we begin by rewriting the function f(x) using the technique of completing the square. This transformation is crucial as it reveals the geometric underpinnings of the problem. We can express x² - 2x + 2 as (x - 1)² + 1 and x² - 4x + 29 as (x - 2)² + 25. Consequently, the function f(x) can be rewritten as: f(x) = √((x - 1)² + 1²) + √((x - 2)² + 5²). This form strongly hints at a geometric interpretation involving distances in the Cartesian plane. The expression √((x - 1)² + 1²) can be interpreted as the distance between a point (x, 0) on the x-axis and the point (1, 1). Similarly, √((x - 2)² + 5²) represents the distance between the point (x, 0) and the point (2, -5). By recognizing these distances, we can reframe the problem as finding the point on the x-axis that minimizes the sum of these two distances. This geometric perspective allows us to utilize geometric principles, such as the triangle inequality and the concept of reflection, to arrive at the solution. This transition from algebraic expressions to geometric representations exemplifies a powerful problem-solving strategy, often leading to elegant and intuitive solutions. Visualizing the problem in this way not only simplifies the solution process but also deepens our understanding of the underlying mathematical concepts.
Geometric Solution Using Reflection
Now that we've established the geometric interpretation, we can employ a clever technique involving reflection to find the minimum value of f(x). Consider the points A(1, 1) and B(2, -5) in the Cartesian plane. The function f(x) represents the sum of the distances from a point P(x, 0) on the x-axis to points A and B. To minimize this sum, we can reflect point A across the x-axis to obtain a new point A'(1, -1). The distance from P to A is equal to the distance from P to A'. Therefore, f(x) = PA + PB = PA' + PB. The sum PA' + PB is minimized when the points A', P, and B are collinear (lie on the same straight line). This is a direct consequence of the triangle inequality, which states that the sum of the lengths of any two sides of a triangle is greater than or equal to the length of the third side. The minimum value of f(x) is then the distance between A' and B, which can be calculated using the distance formula. The distance A'B is given by √((2 - 1)² + (-5 - (-1))²) = √(1² + (-4)²) = √17. Thus, the minimum value m of f(x) is √17. This reflection technique transforms a minimization problem into a simple distance calculation, highlighting the power of geometric intuition in problem-solving. By visualizing the problem geometrically, we can often bypass complex algebraic manipulations and arrive at the solution more efficiently.
Finding the Value of α and Calculating [m] + [α]
Having determined the minimum value m of f(x) to be √17, our next crucial step is to find the value of x (denoted as α) at which this minimum occurs. As established in the previous section, the minimum occurs when the points A'(1, -1), P(α, 0), and B(2, -5) are collinear. This collinearity implies that the slope of the line segment A'P is equal to the slope of the line segment A'B. We can use this condition to determine the value of α. The slope of A'B is calculated as (-5 - (-1)) / (2 - 1) = -4. The slope of A'P is calculated as (0 - (-1)) / (α - 1) = 1 / (α - 1). Setting these slopes equal to each other, we have 1 / (α - 1) = -4. Solving for α, we get 1 = -4(α - 1), which simplifies to 1 = -4α + 4. Further simplification yields 4α = 3, so α = 3/4. Now that we have both m and α, we can calculate [m] + [α]. Since m = √17, we know that 4 < √17 < 5, so [m] = 4. Since α = 3/4, we have [α] = 0. Therefore, [m] + [α] = 4 + 0 = 4. This final calculation demonstrates the seamless integration of geometric insights and algebraic techniques to arrive at the solution. The process involves not only finding the minimum value but also identifying the specific input that produces this minimum, showcasing a comprehensive understanding of the function's behavior.
Conclusion and Summary of Key Concepts
In conclusion, we have successfully determined the least value of the function f(x) = √(x² - 2x + 2) + √(x² - 4x + 29) and the value of x at which this minimum occurs. By employing a combination of algebraic manipulation and geometric interpretation, we found that the minimum value m is √17 and it occurs at x = α = 3/4. Consequently, [m] + [α] = 4. This problem serves as an excellent example of how a seemingly complex analytical challenge can be elegantly solved by leveraging geometric insights. The key steps in our solution were: 1. Completing the square: This allowed us to rewrite the function in a form that suggested a geometric interpretation. 2. Geometric interpretation: Recognizing the expressions as distances in the Cartesian plane. 3. Reflection technique: Reflecting a point across the x-axis to minimize the sum of distances. 4. Using collinearity: Determining the value of α by equating slopes of line segments. 5. Greatest integer function: Calculating [m] + [α] using the greatest integer function. The problem highlights the power of visualizing mathematical concepts and the importance of choosing the right approach. The reflection technique, in particular, is a valuable tool for solving optimization problems involving distances. Understanding these concepts and techniques will undoubtedly enhance your problem-solving skills in calculus and related fields. This exploration not only provides a solution to a specific problem but also reinforces the importance of a holistic approach to mathematical problem-solving, where algebraic and geometric perspectives complement each other to unlock elegant solutions.
Find the least value m of the function f(x) = √(x² - 2x + 2) + √(x² - 4x + 29). If this minimum occurs at x = α, what is the value of [m] + [α], where [.] denotes the greatest integer function?
Minimize f(x) = √(x² - 2x + 2) + √(x² - 4x + 29) - Maxima and Minima Problem Solved
