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MAT2615 ASSIGNMENT 3 2022
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This document contains MAT2615 ASSIGNMENT 3 2022 solutions. Clear step by step calculations and explanations are provided.
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MAT2615ASSIGNMENT 3
2022
,Solution:
𝑦 1
𝑓(𝑥, 𝑦) = − ln(𝑥) + 𝑦 2
𝑥 2
𝑦 1
𝑓𝑥 (𝑥, 𝑦) = − −
𝑥2 𝑥
2𝑦 1
𝑓𝑥𝑥 (𝑥, 𝑦) = +
𝑥3 𝑥2
1
𝑓𝑥𝑦 (𝑥, 𝑦) = −
𝑥2
1
𝑓𝑦 (𝑥, 𝑦) = +𝑦
𝑥
𝑓𝑦𝑦 (𝑥, 𝑦) = 1
𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑝𝑜𝑖𝑛𝑡𝑠:
𝑓𝑥 (𝑥, 𝑦) = 0 𝑎𝑛𝑑 𝑓𝑦 (𝑥, 𝑦) = 0
∴ 𝑓𝑥 (𝑥, 𝑦) = 0
𝑦 1
− − =0
𝑥2 𝑥
−𝑦 − 𝑥 = 0
𝑥+𝑦 =0 1
∴ 𝑓𝑦 (𝑥, 𝑦) = 0
1
+𝑦 =0
𝑥
, 1 + 𝑥𝑦 = 0 2
𝐹𝑟𝑜𝑚 1 𝑥 = −𝑦 𝑎𝑛𝑑 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑖𝑛𝑡𝑜 2
1 + 𝑥𝑦 = 0 2
1 + (−𝑦)𝑦 = 0
1 − 𝑦2 = 0
𝑦2 = 1
𝑦 = ±√1
𝑦 = 1 𝑜𝑟 𝑦 = −1
𝐹𝑜𝑟 𝑦 = 1:
𝑥 = −𝑦
𝑥 = −1
𝐹𝑜𝑟 𝑦 = −1:
𝑥 = −𝑦
𝑥 = −(−1)
𝑥=1
𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑝𝑜𝑖𝑛𝑡𝑠: (−1,1) 𝑎𝑛𝑑 (1, −1)
2
𝐷(𝑎, 𝑏) = 𝑓𝑥𝑥 (𝑎, 𝑏) ∙ 𝑓𝑦𝑦 (𝑎, 𝑏) − [𝑓𝑥𝑦 (𝑎, 𝑏)]
2𝑦 1
𝑓𝑥𝑥 (𝑥, 𝑦) = +
𝑥3 𝑥2
1
𝑓𝑥𝑦 (𝑥, 𝑦) = −
𝑥2
𝑓𝑦𝑦 (𝑥, 𝑦) = 1
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