Word Count : 485

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Let's solve these step by step manually. --- ## Problem (i): $f(x,y) = 4xy + x^4 - y^4$ Step 1: Find stationary points Stationary points occur where the partial derivatives vanish: $$ f_x = \frac{\partial f}{\partial x} = 4y + 4x^3 = 0 $$ $$ f_y = \frac{\partial f}{\partial y} = 4x - 4y^3 = 0 $$ Simplify: 1. $x^3 + y = 0 \implies y = -x^3$ 2. $x - y^3 = 0 \implies x = y^3$ Substitute $y = -x^3$ into $x = y^3$: $$ x = (-x^3)^3 = -x^9 \implies x + x^9 = 0 \implies x(1 + x^8) = 0 $$ * So, $x = 0$ (since $1 + x^8 > 0$ always for real x) * _________ _____ _____ ____ _________ _____ _____ ___ ___ ___ ____ _______.
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