# Advanced Calculus: Second-Order Approximation and the Second-Derivative Test

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1. Let f(x,y) = xy + 2x y - 6xy

(a) Locate the critical points of f(x,y) and determine if they are local maxima, minima, or neither.

(b) Find the first and second order approximations of f(x,y) at the point (1,-1).

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1. Let f(x,y) = xy + 2x y - 6xy

(a) Locate the critical points of f(x,y) and determine if they are local maxima, minima, or neither.

Solution. Since f(x,y) = xy + 2x y - 6xy, we have

So, letting the above two partial derivatives equal to zeros, ,

i.e., .

By discussing if x=0 or y=0, we can get the critical points as follows.

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#### Solution Summary

Critical points and first and second order approximations at a point are found. The solution is detailed.

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