Finding The Minimum Value Of F(x) = √(x² - 2x + 2) + √(x² - 4x + 29)

In this article, we will delve into the problem of finding the minimum value of the function f(x) = √(x² - 2x + 2) + √(x² - 4x + 29). This problem falls under the categories of Algebraic Geometry and Maxima Minima, and it involves dealing with radicals. We will explore a geometric approach to solve this problem, which provides a clear and intuitive understanding of the solution. By carefully analyzing the function, we aim to determine not only the minimum value 'm' but also the value of x (denoted as α) at which this minimum occurs. Furthermore, we will calculate [m] + [α], where [.] represents the greatest integer function. This comprehensive exploration will provide a detailed solution and enhance your understanding of optimization problems involving square roots and quadratic expressions.

The problem at hand requires us to find the least value of the function f(x) = √(x² - 2x + 2) + √(x² - 4x + 29). This is a classic optimization problem that can be approached using various techniques, including calculus and geometric interpretations. The geometric approach is particularly insightful in this case, as it allows us to visualize the problem and find a solution without resorting to complex algebraic manipulations. Our primary goal is to determine the minimum value 'm' of the function and the corresponding value of x, denoted by α, at which this minimum occurs. Additionally, we need to compute [m] + [α], where [.] represents the greatest integer function. To provide a thorough and comprehensive solution, we will first reformulate the function in a geometrically meaningful way. This involves completing the squares inside the square roots and interpreting them as distances in a coordinate plane. By doing so, we transform the problem into a geometric optimization problem, which can be solved using geometric principles. We will then use the properties of straight lines and the triangle inequality to find the minimum distance. This approach not only yields the solution but also provides a deeper understanding of the underlying concepts. We will also discuss the significance of the greatest integer function and its role in the final answer. By breaking down the problem into manageable steps and providing clear explanations, we aim to make the solution accessible and understandable to a wide audience.

To effectively tackle this optimization challenge, we will employ a geometric interpretation that transforms the algebraic problem into a visual one. By rewriting the function f(x) = √(x² - 2x + 2) + √(x² - 4x + 29), we can express it in terms of distances in a coordinate plane. This approach not only simplifies the problem but also provides an intuitive understanding of the solution. Our initial step involves completing the squares inside the square roots, which allows us to rewrite the function in a more geometrically meaningful form. The technique of completing the square is essential for transforming quadratic expressions into a form that can be easily interpreted in terms of distances. By completing the squares, we can rewrite the function as f(x) = √((x - 1)² + (0 - 1)²) + √((x - 2)² + (0 - 5)²). This form immediately suggests that we are dealing with distances between points in a two-dimensional plane. The first term, √((x - 1)² + (0 - 1)²), can be interpreted as the distance between the point (x, 0) and the point (1, 1). Similarly, the second term, √((x - 2)² + (0 - 5)²), represents the distance between the point (x, 0) and the point (2, 5). Thus, the function f(x) represents the sum of the distances from the point (x, 0) on the x-axis to the points A(1, 1) and B(2, 5). This geometric interpretation transforms the original algebraic problem into a geometric optimization problem: finding the point on the x-axis that minimizes the sum of its distances to two fixed points. To solve this geometric problem, we will use the concept of reflection and the properties of straight lines. Reflecting one of the points across the x-axis allows us to transform the problem of minimizing the sum of distances into a problem of finding the shortest path between two points, which is a straight line. The use of reflection is a clever technique that simplifies the problem and provides an elegant solution. By carefully analyzing the geometry of the situation, we can determine the minimum value of the function and the corresponding x-value. This approach not only provides the solution but also deepens our understanding of the connection between algebra and geometry.

Let's consider the points A(1, 1) and B(2, 5) in the coordinate plane. We want to find a point P(x, 0) on the x-axis such that the sum of the distances AP + PB is minimized. To do this, we reflect point A across the x-axis to obtain the point A'(1, -1). The distance AP is equal to A'P, so minimizing AP + PB is equivalent to minimizing A'P + PB. According to the triangle inequality, the shortest distance between two points is a straight line. Therefore, the minimum value of A'P + PB occurs when the points A', P, and B are collinear, meaning they lie on the same straight line. This geometric insight is crucial for solving the problem efficiently. The minimum value of AP + PB is the length of the line segment A'B. We can calculate this distance using the distance formula: A'B = √((2 - 1)² + (5 - (-1))²) = √(1² + 6²) = √37. Therefore, the minimum value of f(x) is √37. To find the value of x at which this minimum occurs, we need to find the x-coordinate of the point P where the line segment A'B intersects the x-axis. The equation of the line A'B can be found using the two-point form: (y - y₁) / (x - x₁) = (y₂ - y₁) / (x₂ - x₁). Substituting the coordinates of A'(1, -1) and B(2, 5), we get: (y - (-1)) / (x - 1) = (5 - (-1)) / (2 - 1) => (y + 1) / (x - 1) = 6 => y + 1 = 6(x - 1) => y = 6x - 7. To find the x-intercept (the point where the line intersects the x-axis), we set y = 0: 0 = 6x - 7 => x = 7/6. Thus, the minimum value of f(x) occurs at x = 7/6. In summary, the minimum value m of f(x) is √37, and it occurs at x = α = 7/6. This geometric approach provides a clear and intuitive way to solve the problem, avoiding complex algebraic manipulations and highlighting the connection between algebra and geometry. The key to this solution lies in the clever use of reflection and the understanding of the triangle inequality.

Now that we have determined the minimum value of the function f(x) and the point at which it occurs, our next step is to calculate [m] + [α], where [.] denotes the greatest integer function. The greatest integer function, also known as the floor function, returns the largest integer less than or equal to the given number. This function is an essential part of many mathematical problems and has various applications in number theory, computer science, and other fields. To find [m] + [α], we first need to evaluate the greatest integer of m and α individually. We found that the minimum value m of f(x) is √37. Since 36 < 37 < 49, we know that √36 < √37 < √49, which means 6 < √37 < 7. Therefore, the greatest integer of √37, denoted as [√37], is 6. This calculation is straightforward and relies on our understanding of square roots and inequalities. Next, we need to find the greatest integer of α. We found that the minimum value of f(x) occurs at x = α = 7/6. To find the greatest integer of 7/6, we can perform the division: 7 ÷ 6 = 1 with a remainder of 1. This means that 7/6 is equal to 1 and 1/6, which is between 1 and 2. Therefore, the greatest integer of 7/6, denoted as [7/6], is 1. Now that we have calculated [m] = 6 and [α] = 1, we can simply add them together to find the final answer. The sum [m] + [α] is equal to 6 + 1, which is 7. Thus, the value of [m] + [α] is 7. This final calculation is straightforward and completes the solution to the problem. The greatest integer function is a fundamental concept in mathematics, and its proper application is crucial for solving problems involving integers and real numbers. By carefully evaluating the greatest integer of both m and α, we arrive at the correct answer. This comprehensive step-by-step solution demonstrates the importance of each component in the problem and provides a clear path to the final answer.

In conclusion, we have successfully found the least value of the function f(x) = √(x² - 2x + 2) + √(x² - 4x + 29) and the corresponding value of x. Through the geometric interpretation, we transformed the problem into minimizing the sum of distances from a point on the x-axis to two fixed points in the coordinate plane. By reflecting one of the points across the x-axis and applying the triangle inequality, we determined that the minimum value m is √37, which occurs at x = α = 7/6. Furthermore, we calculated [m] + [α], where [.] represents the greatest integer function. We found that [√37] = 6 and [7/6] = 1, so [m] + [α] = 6 + 1 = 7. Therefore, the final answer to the problem is 7, which corresponds to option (B). This problem serves as an excellent example of how geometric interpretations can simplify complex algebraic problems and provide intuitive solutions. The geometric approach not only allowed us to find the minimum value but also provided a deeper understanding of the function's behavior. By carefully applying the concepts of reflection, the triangle inequality, and the greatest integer function, we were able to solve the problem efficiently and accurately. The combination of algebraic manipulation and geometric reasoning is a powerful tool in problem-solving, and this example highlights the importance of developing both skills. This detailed solution provides a comprehensive understanding of the problem and its solution, which will be valuable for anyone studying optimization problems and related topics in mathematics.