H – Quadratic Optimisation | H – 二次优化

📚 H – Quadratic Optimisation | H – 二次优化

Secondary school mathematics often involves finding the best possible outcome—maximum profit, minimum cost, or optimal dimensions—and when the relationship is quadratic, we step into quadratic optimisation. This topic equips you to locate the peak or trough of a parabola and apply it to real-world constraints.

中学数学常常涉及寻找最佳结果——最大利润、最小成本或最优尺寸——当这种关系是二次关系时，我们就进入了二次优化的领域。这一主题使你能够定位抛物线的顶点，并将其应用于现实世界的约束条件中。

1. Introduction to Quadratic Optimisation | 二次优化导论

Quadratic optimisation is the process of maximising or minimising a quadratic function, possibly subject to constraints. In IB Mathematics, this appears in contexts ranging from pure vertex problems to real-life scenarios like maximising area or minimising surface area. The quadratic function’s graph is a parabola, and its extremum (maximum or minimum) occurs at the vertex.

二次优化是最大化或最小化一个二次函数的过程，通常可能受到约束条件的限制。在IB数学中，这出现在从纯顶点问题到实际场景（如最大化面积或最小化表面积）的各种背景中。二次函数的图像是一条抛物线，其极值（最大值或最小值）发生在顶点处。

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2. Standard Form of a Quadratic Function | 二次函数的标准形式

The general quadratic function is expressed as f(x) = ax² + bx + c, where a ≠ 0. The coefficient a determines the direction of opening: if a > 0, the parabola opens upwards and has a minimum; if a < 0, it opens downwards and has a maximum.

一般二次函数表示为 f(x) = ax² + bx + c，其中 a ≠ 0。系数 a 决定了开口方向：如果 a > 0，抛物线开口向上，有最小值；如果 a < 0，开口向下，有最大值。

f(x) = ax² + bx + c

The y-intercept is (0, c), and the axis of symmetry is the vertical line x = -b/(2a). Understanding these elements is key to optimisation.

y轴截距是 (0, c)，对称轴是垂直线 x = -b/(2a)。理解这些要素是优化的关键。

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3. Vertex Form and Completing the Square | 顶点形式与配方法

By completing the square, we rewrite f(x) = ax² + bx + c as f(x) = a(x – h)² + k, where (h, k) is the vertex. The value h = -b/(2a) and k = f(h) = c – b²/(4a). This form directly reveals the extremum: k is the minimum when a > 0 or maximum when a < 0.

通过配方法，我们可以将 f(x) = ax² + bx + c 重写为 f(x) = a(x – h)² + k，其中 (h, k) 是顶点。h = -b/(2a)，k = f(h) = c – b²/(4a)。这种形式直接揭示了极值：当 a > 0 时，k 是最小值；当 a < 0 时，k 是最大值。

f(x) = a(x – h)² + k,   h = -b/(2a),   k = c – b²/(4a)

Completing the square is a fundamental technique for quadratic optimisation without calculus, and it also helps when handling constraints.

配方法是二次优化中无需微积分的基本技巧，也在处理约束条件时有所帮助。

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4. Unconstrained Optimisation: Finding the Vertex | 无约束优化：求顶点

If no restrictions are placed on x, the global optimum is at the vertex. For a > 0, the minimum value is k; for a < 0, the maximum value is k. The x-coordinate h = -b/(2a) gives the optimal input.

如果没有对 x 施加限制，全局最优解就在顶点处。对于 a > 0，最小值是 k；对于 a < 0，最大值是 k。顶点的 x 坐标 h = -b/(2a) 给出了最优输入值。

Example: Maximise f(x) = -2x² + 8x – 3. Here a = -2 < 0, so maximum occurs at h = -8/(2×(-2)) = 2. Then k = f(2) = -2(4) + 16 - 3 = 5. Maximum value is 5 at x = 2.

例：最大化 f(x) = -2x² + 8x – 3。这里 a = -2 < 0，所以最大值在 h = -8/(2×(-2)) = 2。然后 k = f(2) = -2(4) + 16 - 3 = 5。最大值为5，在 x = 2 处。

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5. The Nature of Turning Points (Maximum/Minimum) | 转折点的性质（最大值/最小值）

The vertex is a turning point. The second derivative (if using calculus) confirms concavity: f”(x) = 2a. If a > 0, f” > 0 implies a local minimum; if a < 0, f'' < 0 implies a local maximum. For quadratics, the local extremum is also global.

顶点是一个转折点。二阶导数（如果使用微积分）可以确认凹性：f”(x) = 2a。如果 a > 0，f” > 0 意味着局部最小值；如果 a < 0，f'' < 0 意味着局部最大值。对于二次函数，局部极值也是全局极值。

When the domain is all real numbers, the range is [k, ∞) for a > 0 or (-∞, k] for a < 0. This range understanding helps in constrained problems where you check whether the vertex lies within the allowed interval.

当定义域为所有实数时，值域对于 a > 0 是 [k, ∞)，对于 a < 0 是 (-∞, k]。对值域的理解有助于在约束问题中检查顶点是否位于允许的区间内。

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6. Domain Restrictions and Feasible Regions | 定义域限制与可行区域

In many optimisation problems, x cannot be any real number—it may be limited to a specific interval [p, q] or by practical constraints (e.g., positive lengths, non-negative quantities). The optimal value may then occur at a boundary rather than the vertex.

在许多优化问题中，x 不能是任意实数——它可能被限制在一个特定区间 [p, q] 或受实际约束（例如正长度、非负量）。此时最优值可能出现在边界处，而不是顶点。

We must evaluate f(x) at the vertex h (if h lies in the interval) and at the endpoints p and q. The largest/smallest among these is the constrained optimum.

我们必须计算顶点 h 处的 f(x)（如果 h 在区间内），以及端点 p 和 q 处的值。这些值中的最大/最小值就是约束下的最优解。

Example: Minimise f(x) = x² – 4x + 5 over 0 ≤ x ≤ 3. Vertex at x = 2, f(2)=1. Endpoints: f(0)=5, f(3)=2. Minimum is 1 (vertex inside); maximum is 5 (endpoint).

例：在 0 ≤ x ≤ 3 上最小化 f(x) = x² – 4x + 5。顶点在 x = 2，f(2)=1。端点：f(0)=5，f(3)=2。最小值是1（顶点在内）；最大值是5（端点）。

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7. Linear Constraints and Substitution Method | 线性约束与代入法

A common IB style problem is: ‘Given a linear relation between two variables, optimise a quadratic expression.’ For instance, x + y = 10, find the minimum of x² + y². Use the constraint to express y = 10 – x, then substitute to obtain a single-variable quadratic: q(x) = x² + (10 – x)² = 2x² – 20x + 100. Then find vertex.

IB 常见的题型是：“给定两个变量之间的线性关系，优化一个二次表达式。”例如，x + y = 10，求 x² + y² 的最小值。利用约束条件将 y 表示为 y = 10 – x，然后代入得到单变量二次函数：q(x) = x² + (10 – x)² = 2x² – 20x + 100。然后求顶点。

This substitution method reduces a two-variable problem to a one-variable quadratic optimisation. Always check the feasible domain from the constraint (e.g., if x and y must be non-negative, then x ∈ [0,10]).

这种代入法将双变量问题转化为单变量二次优化。始终根据约束检查可行域（例如，若 x 和 y 必须为非负，则 x ∈ [0,10]）。

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8. Optimisation with Multiple Constraints | 多约束下的优化

Sometimes several inequalities define a feasible region, such as x ≥ 0, y ≥ 0, and 2x + y ≤ 20. The objective might be a quadratic function P = x² + y. The process involves identifying candidate points: vertices of the feasible polygon and any interior stationary point of the objective (if it lies inside). For a quadratic objective, the interior extremum is found by completing the square after substitution if a linear relation allows.

有时多个不等式定义了可行区域，如 x ≥ 0, y ≥ 0 且 2x + y ≤ 20。目标可能是二次函数 P = x² + y。过程包括确定候选点：可行多边形的顶点以及目标函数的任何内部驻点（如果它位于内部）。对于二次目标，如果线性关系允许，可以通过代入后配方法来找到内部极值。

In a fully linear constraint set with a quadratic objective, check boundaries by expressing y in terms of x along each edge and optimising the resulting quadratic on an interval.

在线性约束集且二次目标的情况下，通过沿每条边将 y 表示为 x 的函数，并在一个区间上优化得到的二次函数，来检查边界。

This also links to the concept of convex functions: if a > 0, the function is convex, and the minimum over a convex polygon occurs either at a vertex or along an edge if the unconstrained minimum lies outside.

这也与凸函数的概念相关：如果 a > 0，函数是凸的，那么在一个凸多边形上的最小值要么出现在顶点，要么如果无约束最小值在外部，则沿着一条边出现。

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9. Application: Maximising Area | 应用：面积最大化

Classic problem: A farmer has 100 m of fencing to enclose a rectangular area against a barn (so only three sides need fencing). Find dimensions for maximum area. Let the side parallel to barn be x, and the two adjacent sides be y. Then x + 2y = 100, area A = xy. Substitute x = 100 – 2y → A(y) = (100 – 2y)y = 100y – 2y². This is a quadratic in y with a = -2, maximum at y = -100/(2×(-2)) = 25, so x = 50. Max area = 1250 m².

经典问题：一位农民有100米长的围栏，想靠着一个谷仓围一个矩形区域（因此只需围三面）。求最大面积时的尺寸。设平行于谷仓的边为 x，两条相邻边为 y。则 x +

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