Word Count : 490

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We are to locate and classify the stationary points of the given functions manually. We will compute partial derivatives, solve for stationary points, and classify using the second partial derivative test. --- ### (i) $f(x,y) = 4xy + x^4 - y^4$ #### Step 1: First-order partial derivatives $$ f_x = \frac{\partial f}{\partial x} = 4y + 4x^3 $$ $$ f_y = \frac{\partial f}{\partial y} = 4x - 4y^3 $$ Set both to zero for stationary points: 1. $4y + 4x^3 = 0 \implies y = -x^3$ 2. $4x - 4y^3 = 0 \implies x = y^3$ #### Step 2: Solve the system Substitute $y = -x^3$ into $x = y^3$: $$ x = (-x^3)^3 \implies x = -x^9 $$ $$ x + _______ ___ ______ ________ ____ ___ _______ ______.
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