Word Count : 363

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We are to find the maximum and minimum values of the function $$ f(x, y) = xy $$ subject to the constraint (ellipse): $$ x^2 + 4y^2 = 4 $$ --- ### Step 1: Define Lagrangian Let the constraint be: $$ g(x, y) = x^2 + 4y^2 - 4 = 0 $$ Form the Lagrangian: $$ \mathcal{L}(x, y, \lambda) = xy - \lambda(x^2 + 4y^2 - 4) $$ --- ### Step 2: Take partial derivatives and set to 0 We compute the gradient equations: 1. $\frac{\partial \mathcal{L}}{\partial x} = y _____ _______ ____ _____ ____.
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