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We are to find the extreme values of $$ f(x,y) = x^2 + y $$ subject to $$ x^2 + 2y^2 = 1 $$ using manual numerical calculation (without Python). --- ### Step 1: Use Lagrange multipliers We want to extremize $f(x,y) = x^2 + y$ subject to $g(x,y) = x^2 + 2y^2 - 1 = 0$. The method of Lagrange multipliers gives: $$ \nabla f = \lambda \nabla g $$ Compute gradients: $$ \nabla f = (2x, _____ ______ _________ __________ _____ _____ _______ _______.
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