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

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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.

...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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