Word Count : 349

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We are asked to find the extreme values of (f(x, y) = x^2 + y) subject to the constraint (x^2 + 2y^2 = 1). Use the method of Lagrange multipliers. Let (\lambda) be the multiplier. Then we solve: [ \nabla f = \lambda \nabla g, \quad g(x, y) = x^2 + 2y^2 - 1 = 0 ] 1. Compute gradients: [ \nabla f = (2x, 1), \quad \nabla g = (2x, 4y) ] So the Lagrange equations are: [ 2x = \lambda 2x \quad \Rightarrow \quad 2x ____ ____ ____ _____ ____ _____ ________.
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