(a)5%)let fx, y) = x^4 + y^4 - 4xy + 1. and classify each critical point find all critical points of fx, y) as a local minimum, local maximum or saddle point.

has critical points wherever the partial derivatives vanish:

Then

If , then ; critical point at (0, 0)If , then ; critical point at (1, 1)If , then ; critical point at (-1, -1)

has Hessian matrix

with determinant

At (0, 0), the Hessian determinant is -16, which indicates a saddle point.At (1, 1), the determinant is 128, and , which indicates a local minimum.At (-1, -1), the determinant is again 128, and , which indicates another local minimum.

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