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Lagrange Multipliers- HW Problems
Find the absolute maximum and minimum values of the following
functions subject to the given constraints.
1. 𝑓(𝑥, 𝑦) = 𝑦 subject to 9𝑥 2 + 4𝑦 2 = 36
2. 𝑓(𝑥, 𝑦) = 𝑥 3 + 𝑦 3 subject to 𝑥 2 + 𝑦 2 ≤ 1
3. 𝑓(𝑥, 𝑦) = 𝑥 2 + 2𝑦 2 subject to 𝑥 2 + 𝑦 2 = 1
4. 𝑓(𝑥, 𝑦) = 𝑥 2 + 2𝑦 2 subject to 𝑥 2 + 𝑦 2 ≤ 1
5. 𝑓(𝑥, 𝑦, 𝑧) = 𝑥 + 𝑧 subject to 𝑥 2 + 𝑦 2 + 2𝑧 2 = 1
6. 𝑓(𝑥, 𝑦, 𝑧) = 𝑥 2 + 𝑦 2 + 𝑧 2
subject to (𝑥 − 1)2 + (𝑦 − 2)2 + (𝑧 + 2)2 ≤ 36.
7. A rectangular box without a top is to be made from 24 𝑓𝑡 2 of
material. Find the maximum volume of the box.
Lagrange Multipliers- HW Problems
Find the absolute maximum and minimum values of the following
functions subject to the given constraints.
1. 𝑓(𝑥, 𝑦) = 𝑦 subject to 9𝑥 2 + 4𝑦 2 = 36
2. 𝑓(𝑥, 𝑦) = 𝑥 3 + 𝑦 3 subject to 𝑥 2 + 𝑦 2 ≤ 1
3. 𝑓(𝑥, 𝑦) = 𝑥 2 + 2𝑦 2 subject to 𝑥 2 + 𝑦 2 = 1
4. 𝑓(𝑥, 𝑦) = 𝑥 2 + 2𝑦 2 subject to 𝑥 2 + 𝑦 2 ≤ 1
5. 𝑓(𝑥, 𝑦, 𝑧) = 𝑥 + 𝑧 subject to 𝑥 2 + 𝑦 2 + 2𝑧 2 = 1
6. 𝑓(𝑥, 𝑦, 𝑧) = 𝑥 2 + 𝑦 2 + 𝑧 2
subject to (𝑥 − 1)2 + (𝑦 − 2)2 + (𝑧 + 2)2 ≤ 36.
7. A rectangular box without a top is to be made from 24 𝑓𝑡 2 of
material. Find the maximum volume of the box.
