Minimum, Maximum, Saddle point of f(x,y)=xy(1−9x−3y)?

[匿名]

2012-01-25 6:52 am

The function f(x,y)=xy(1−9x−3y) has 4 critical points. List them and specify whether they are minimum, maximum or a saddle point.

Points should be entered as ordered pairs and listed in increasing lexicographic order. By that we mean that (x,y) comes before (z,w) if x<z or if x=z and y<w.

What is lexicographic order?

Please show steps. Thank you very much!

回答 (1)

2012-01-25 7:20 am

✔ 最佳答案

For the critical points, set the first partial derivatives of f equal to 0.

Since f(x,y) = xy - 9x^2 y - 3xy^2,
f_x = y - 18xy - 3y^2
f_y = x - 9x^2 - 6xy.

Setting these equal to 0:
y(1 - 18x - 3y) = 0
x(1 - 9x - 6y) = 0.

Solving yields (x, y) = (0, 0), (0, 1/3), (1/9, 0), or (1/27, 1/9).
(This is in lexicographic order; compare the first entries; otherwise if they're equal, compare the second entries; it's essentially how one puts words in alphabetical order.)

To classify these points, use the Second Derivative Test.
f_xx = -18y, f_yy = -6x, f_xy = 1 - 18x - 6y
==> D = (f_xx)(f_yy) - (f_xy)^2 = 108xy - (1 - 8x - 6y)^2.

(i) Since D(0, 0), D(0, 1/3), D(1/9, 0) < 0,
(0, 0), (0, 1/3), (1/9, 0) are all saddle points.

(ii) Since D(1/27, 1/9) > 0, and f_xx (1/27, 1/9) < 0,
there is a local maximum at (1/27, 1/9).

I hope this helps!

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