📚 Quadratic Optimisation: Maximising and Minimising Quadratic Functions | 二次优化：二次函数的最大化与最小化
In IB Mathematics, quadratic optimisation involves finding the maximum or minimum value of a quadratic function, which frequently appears in real-world contexts such as maximising area, profit, or minimising cost. Understanding how to determine the optimal value and the corresponding variable is essential for both Analysis and Approaches and Applications and Interpretation courses. This article systematically covers techniques from completing the square to constrained optimisation, equipping you with strategies for examination success.
在IB数学中，二次优化涉及寻找二次函数的最大值或最小值，这些问题常出现在实际场景中，如面积最大化、利润最大化或成本最小化。掌握如何确定最优值及对应的变量对分析与方法和应用与解释课程都至关重要。本文系统地介绍从配方法到约束优化的各种技巧，助你掌握应试策略。
1. Introduction to Quadratic Optimisation | 二次优化简介
where a, b, c are constants and a ≠ 0. The graph is a parabola. If a > 0, the parabola opens upward and has a minimum point; if a < 0, it opens downward and has a maximum point. The optimisation problem is to find the value of x that gives this extreme value, and to compute that extreme value itself.
二次优化是指最大化或最小化二次目标函数的过程，通常受一定约束。单变量二次函数的一般形式为f(x) = ax² + bx + c，其中a、b、c为常数且a ≠ 0。图像是一条抛物线。若a > 0，抛物线开口向上，有最小值点；若a < 0，开口向下，有最大值点。优化问题就是要找出使函数取得极值的x值，并计算该极值本身。
In IB examinations, problems can be unconstrained (simply find the vertex) or constrained (e.g., with a linear relationship between variables, or a restricted domain). Both algebraic and graphical methods are applicable. Key techniques include finding the axis of symmetry, completing the square to obtain vertex form, and using the discriminant for related optimisation problems.
在IB考试中，问题可以是无约束的（直接求顶点），也可以是带约束的（例如变量间存在线性关系，或定义域受限）。代数方法和图像方法均可使用。关键技巧包括求对称轴、配方法得到顶点形式，以及利用判别式解决相关优化问题。
2. Standard Form of a Quadratic Function | 二次函数的标准形式
ediately reveals the y-intercept (0, c) and the coefficient a which determines the direction of opening. However, the vertex is not directly visible. To locate the optimal point, we can use the formula for the x-coordinate of the vertex.
标准形式f(x) = ax² + bx + c可直接显示y轴截距(0, c)以及决定开口方向的系数a。但顶点并不直接可见。要找到最优点，可使用顶点的x坐标公式。
x = -b/(2a)
This formula arises from completing the square or from calculus by setting the derivative to zero. It is valid for any quadratic function, provided a ≠ 0. Then the optimal value is f(-b/(2a)).
该公式由配方法或微积分中令导数为零得出。它对任何二次函数均成立，只要a ≠ 0。最优值即为f(-b/(2a))。
= 2, b = -8, so x = -(-8)/(2×2) = 8/4 = 2. The minimum value is f(2) = 2(4) - 16 + 5 = -3.
示例：对于f(x) = 2x² - 8x + 5，a = 2, b = -8，故x = -(-8)/(2×2) = 8/4 = 2。最小值为f(2) = 2×4 - 16 + 5 = -3。
3. Vertex Form and Completing the Square | 顶点形式与配方法
are transforms f(x) = ax² + bx + c into the vertex form f(x) = a(x - h)² + k, where (h, k)
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