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第 15 週 · 不等式與線性規劃 · 第 6 節

線性規劃應用題（角點法）＋ 本週關卡

在可行域內，讓某個量 P = ax + by 最大或最小 —— 這就是線性規劃。核心一句話：最優值一定在可行域的「頂點（角點）」取得，用角點法四步就能秒解。本節為 W15 收官，含整週 BOSS 總關卡。

▶ 第一關： 線性規劃是什麼 + 角點定理
xyfeasible region(0,0)vertexvertexvertexmaximise P = ax + byanswer sits at a corner
corner-point theorem: optimum is at a vertexoptimum vertexP increases →feasibleequal-P lines (parallel)
check ALL vertices of P = 3x + 2y(0, 0)3·0 + 2·0 = 0(4, 0)3·4 + 2·0 = 12 ◀ max(0, 4)3·0 + 2·4 = 8(2, 2)3·2 + 2·2 = 10
▶ 第二關： 角點法四步（求頂點 → 代入比大小）
1. modeldraw region2. verticessolve lines3. sub in Peach vertex4. comparemax / mincorner-point method: 4 steps
vertex = intersection of two boundary linesxyx + y = 6x = 2(2, 4)42
vertex → P table (compare, then pick best)vertexP = 4x + yP = 2x + 5y(0,0)00(5,0)20 ◀10(0,3)315(3,2)1416 ◀different P → different best vertex
▶ 第三關： 生產應用題（求最大利潤）
production: x of A, y of B, profit P = 30x + 20yx (A)y (B)(0,0)(40,0)(30,20) best(0,50)feasible
profit at each vertex, P = 30x + 20y(0, 0)0 + 0 = $0(40, 0)1200 + 0 = $1200(0, 50)0 + 1000 = $1000(30, 20)900 + 400 = $1300 ◀
⚔ BOSS： W15 本週總關卡（核心主題綜合）
W15 knowledge map: number line → plane optimisationD1 linear inequalitiesD2 quadratic (graph)D3 quadratic (algebra)D4 half-planeD5 feasible regionD6 linear programmingcorner-point methodeverything feeds into the corner-point method
BOSS demo: maximise P = x + 3y on the regionxy(0,0)(6,0)(4,2)(0,4) maxP: 0, 6, 10, 12 → 12best at (0,4)