Word Count : 651

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आपके द्वारा दिये गए प्रश्न में दो विभिन्न गणितीय फलनों के स्तब्ध बिन्दुओं (critical points) को निर्धारित और उनका वर्गीकरण (classification) करने के लिए हम चरणबद्ध तरीके से इसका समाधान करेंगे। ### (i) \( f(x, y) = 4xy + x^4 - y^4 \) 1. पहले आंशिक अवकलज (Partial Derivatives): - \( f_x(x, y) = \frac{\partial}{\partial x}(4xy + x^4 - y^4) = 4y + 4x^3 \) - \( f_y(x, y) = \frac{\partial}{\partial y}(4xy + x^4 - y^4) = 4x - 4y^3 \) 2. स्तब्ध बिन्दु (Critical Points) के लिए समीकरण हल करें: - \( f_x(x, y) = 0 \Rightarrow 4y + 4x^3 = 0 \Rightarrow y = -x^3 \) - \( f_y(x, y) = 0 \Rightarrow 4x - 4y^3 = 0 \Rightarrow x = y^3 \) अब, \( y = -x^3 __________ ______ _______ _____ ____ __________ ______ _____ _______ ____ ______ ___.
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