📚 IB Mathematics: Optimisation | IB数学：优化
Optimisation is a cornerstone of applied calculus, allowing us to determine the best possible outcome — be it maximum profit, minimum surface area, or shortest time — under given constraints. In IB Mathematics, particularly the Analysis and Approaches and Applications and Interpretation courses, you will encounter optimisation problems that require setting up a function of a single variable and then using differentiation to find its extreme values.
优化是应用微积分的基石，它使我们能够在给定约束条件下确定最佳结果——无论是最大利润、最小表面积还是最短时间。在IB数学中，尤其是分析与方法和应用与解释课程中，你会遇到需要建立单变量函数、再通过求导找到极值的优化问题。
Mastering optimisation involves a clear sequence: define the quantity to be optimised, express it as a function of one variable using given conditions, find the domain, differentiate, locate stationary points, and finally classify them using derivative tests or boundary considerations. This article breaks down the entire process into digestible sections, complete with geometrical and economic contexts to mirror typical IB exam questions.
掌握优化需要一套清晰的流程：确定要优化的量，利用已知条件将其表示为单变量函数，找出定义域，求导，定位驻点，最后用导数检验或边界考量进行分类。本文将整个过程分解为易于理解的小节，并辅以几何与经济背景，与典型的IB考题相呼应。
1. What Is Optimisation? | 什么是优化？
In calculus, optimisation refers to finding the maximum or minimum values of a function — the global optimum — ints where f(x) attains its highest or lowest value, either over all real numbers or on a specific closed interval [a, b].
在微积分中，优化指的是在可行集合内找出函数的最大值或最小值——全局最优值。对于函数 y = f(x)，我们通常关心 f(x) 取得最高值或最低值的点，无论是在全体实数上还是在特定的闭区间 [a, b] 上。
IB problems typically present a real-life scenario: a manufacturer wants to minimise the cost of a cylindrical can, or a farmer needs to maximise the area of a rectangular field bordering a river. Translating these scenarios into a mathematical function is the first and most critical step.
IB考题通常会给出一个实际场景：制造商希望最小化圆柱罐的成本，或农民想最大化靠河矩形田地的面积。将这些场景转化为数学函数是第一步也是最关键的一步。
2. The Principle of Stationary Points | 驻点原理
A stationary point occurs where the first derivative is zero or undefined, i.e., f'(x) = 0 or f'(x) does not exist. For a smooth, differentiable function, solving f' oordinates of potential maxima, minima, or points of inflection with a horizontal tangent.
驻点出现在一阶导数为零或不存在处，即 f'(x) = 0 或 f'(x) 不存在。对于光滑、可微的函数，解方程 f'(x) = 0 可得到潜在的极大值点、极小值点或具有水平切线的拐点的 x 坐标。
tand that not every stationary point is an optimum: a horizontal point of inflection has f' n does not change from increasing to decreasing, so it is neither a maximum nor a minimum.
必须明白并非每个驻点都是最优值：水平拐点处的导数为零，但函数没有从递增变为递减，因此既不是极大值也不是极小值。
3. First Derivative Test | 一阶导数检验
The first derivative test examines the sign of f'(x) to the left and right of a stationary point x = c. If f' hanges from positive to negative as x passes through c, then f(c) is a local maximum. If f' f(c) is a local minimum. If there is no sign change, the point is a stationary inflection.
一阶导数检验考察驻点 x = c 左右两侧 f'(x) 的正负号。如果在 x 通过 c 时 f'(x) 由正变负，那么 f(c) 是局部极大值。如果 f'(x) 由负变正，则 f(c) 是局部极小值。如果没有符号变化，该点是驻点拐点。
To perform the test, pick a value slightly less than c and one slightly greater, then evaluate f'(x). This method is particularly useful when the second derivative is difficult to compute or equal to zero.
进行检验时，选取一个略小于 c 的值和一个略大于 c 的值，计算 f'(x)。当二阶导数难以计算或等于零时，此方法尤为有用。
4. Second Derivative Test | 二阶导数检验
If f''(x) exists at a stationary point c where f'(c) = 0, the sign of the second derivative gives instant classification: if f''(c) > 0, the function is concave up and f(c) is a local minimum; if f''(c) < 0, the function is concave down and f(c) is a local maximum.
若驻点 c 处存在二阶导数且 f'(c) = 0，则二阶导数的符号可立即进行分类：若 f''(c) > 0，函数是凹向上的，f(c) 为局部极小值；若 f''(c) < 0，函数是凹向下的，f(c) 为局部极大值。
When f'' point could be a maximum, a minimum, or an inflection. In such cases, revert to the first derivative test or examine higher-order derivatives.
当 f''(c) = 0 时，该检验无法确定；该点可能是极大值、极小值或拐点。这种情况下，应退回到一阶导数检验或考察更高阶导数。
| Second Derivative Sign | Concavity | Stationary Point Type |
|---|---|---|
| f''(c) > 0 | Concave up ∪ | Local minimum |
| f''(c) < 0 | Concave down ∩ | Local maximum |
| f''(c) = 0 | Undetermined | Need further analysis |
5. Closed Interval Method | 闭区间法
When optimising a continuous function on a closed interval [a, b], the absolute maximum and minimum must occur either at a stationary point inside the interval or at the endpoints. The procedure is: find all critical numbers in (a, b), evaluate f at these points and at a and b, then compare the values.
在闭区间 [a, b] 上优化一个连续函数时，绝对最大值和最小值必然出现在区间内部的驻点或端点处。步骤是：找出 (a, b) 内所有临界值，计算 f 在这些点以及 a 和 b 处的值，然后比较大小。
This method guarantees finding the global extrema on the interval without needing to classify each critical point individually. It is highly favoured in IB problems where a specific domain is naturally imposed by physical constraints, such as non-negative lengths.
该方法能保证找到区间上的全局极值，无需对每个临界点逐一分类。在IB问题中，由于物理约束（例如长度非负）往往自然规定了一个特定定义域，这种方法备受青睐。
6. Modelling with Optimisation | 优化建模
Modelling is the art of converting a wordy problem into a mathematical function. Start by identifying the quantity to be maximised or minimised — area, volume, cost, distance — and assign a suitable variable. Then use the given constraints to express this quantity as a function of a single variable, usually by eliminating one variable using a constraint equation.
建模是将繁琐的文字问题转化为数学函数的艺术。首先确定要最大化或最小化的量——面积、体积、成本、距离——并指定合适的变量。然后利用给定约束将这个量表示为单变量函数，通常通过约束方程消去一个变量。
- x, so A(x) = x[(P/2) - x]. This reduction is the heart of single-variable optimisation.
例如，要求设计一个周长固定为 P 且面积 A 最大的矩形围栏，设边长为 x 和 y。约束条件 2x + 2y = P 得出 y = (P/2) - x，因此 A(x) = x[(P/2) - x]。这种降元是单变量优化的核心。
7. Geometric Optimisation Problems | 几何优化问题
Geometric optimisation frequently appears in IB exams: maximising the volume of a box made from a flat sheet by cutting squares from the corners, minimising the surface area of a cylinder for a fixed volume, or finding the dimensions of the largest rectangle inscribed in a semicircle. All demand a solid grasp of geometric formulas and their differentiation.
几何优化在IB考试中频繁出现：从平板上切去四角做成盒子的体积最大化、固定体积下圆柱表面积最小化，或找出内接于半圆的最大矩形的尺寸。所有这些都需要牢固掌握几何公式及其微分。
and checking the feasible domain 0 < x < 10 yields the optimal cut size for maximum volume.
考虑从一个 20 cm × 30 cm 的薄板四角各切去边长为 x 的正方形后制成无盖盒。体积函数为 V(x) = x(30 - 2x)(20 - 2x)。求导，解 dV/dx = 0，并在可行域 0 < x < 10 内检验，即可得到使体积最大的最优切角尺寸。
8. Economic Optimisation Problems | 经济优化问题
Economics provides rich optimisation scenarios: maximising profit, minimising average cost, or finding the production level where marginal cost equals marginal revenue. In IB, these problems are often framed as polynomial, rational, or exponential functions, and you will be asked to find the quantity that yields the extreme value.
经济学提供了丰富的优化场景：利润最大化、平均成本最小化，或找到边际成本等于边际收入的生产水平。在IB中，这些问题常以多项式、有理函数或指数函数为框架，要求找出令目标量取得极值的数量。
If profit Π(x) = R(x) - C(x), where x is units produced, then setting the derivative Π'(x) = 0 gives the critical points. A typical step is to confirm that Π'' t the candidate point to ensure a maximum. Be mindful that x must be a meaningful quantity — often an integer — so rounding to the nearest feasible unit may be required.
若利润 Π(x) = R(x) - C(x)，其中 x 为产量，则令 Π'(x) = 0 可得临界点。典型步骤是在候选点处确认 Π''(x) < 0 以保证极大值。注意 x 必须是有意义的量——通常是整数——因此可能需要就近舍入到可行单位。
9. Constrained Optimisation by Substitution | 代入法约束优化
onstraint x² + y² = D². Solve for y in terms of x (positive branch) and substitute into A.
当优化问题涉及两个由约束方程关联的变量时，最直接的IB方法是用代入法将函数降为单变量。例如，要在固定对角线 D 下最大化矩形面积，可写出 A = xy 并受限于 x² + y² = D²。将 y 用 x 表示（取正值支）并代入 A。
This method can be applied to more complex relationships, including those involving trigonometric identities or exponential forms. Substitution keeps the mathematics within single-variable calculus, which is fully aligned with the IB syllabus at both Standard and Higher Level, avoiding the need for Lagrange multipliers.
该方法可应用于更复杂的关系，包括包含三角恒等式或指数形式的关系。代入法将数学处理保持在单变量微积分内，完全符合IB标准水平和高级水平的大纲要求，无需使用拉格朗日乘子。
10. Common Pitfalls and Pro Tips | 常见陷阱与专业技巧
A frequent mistake is forgetting to check the domain of the function. Optimisation variables often represent lengths, areas, or quantities that cannot be negative or exceed physical limits. Always explicitly state the feasible domain before differentiating, and test endpoints if the domain is closed.
一个常见错误是忘记检查函数的定义域。优化变量通常代表长度、面积或数量，它们不能为负或超出物理限制。求导前务必明确给出可行定义域，并在闭区间上检验端点。
Another pitfall is misinterpreting the second derivative test when f''(c) = 0. Do not assume it is an inflection; use a sign chart of f' always answer the original question: the value of x that optimises is only half the answer — orresponding optimum value (maximum volume, minimum cost, etc.) and include correct units.
另一陷阱是当 f''(c) = 0 时误用二阶导数检验。切勿直接假设它是拐点；应改用 f'(x) 的符号表。此外，始终要回答原始问题：最优 x 值只是一半答案——你必须计算出对应的最优值（最大体积、最小成本等）并包含正确单位。
Finally, diagram sketching where possible strengthens your understanding. Label variables, constraints, and the objective function. A clear diagram not only reduces algebraic errors but also impresses examiners with a well-structured solution.
最后，尽可能绘制简图可加深理解。标出变量、约束和目标函数。清晰的图示不仅能减少代数错误，还能以结构清晰的解答给阅卷人留下良好印象。
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