4. +-/16 points scalcCC4117.009.MI. My Notes Find the local maximum and minimum

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4. +-/16 points scalcCC4117.009.MI. My Notes Find the local maximum and minimum values and saddle point(s) of the function. If you have three dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter NONE in any unused answer blanks.) rx, y) =x3-27xy + 27y3 maximum (smaller x value) (larger x value) minimum (smaller x value) (larger x value) saddle points (smallest x value) ) (largest x value) Need Help?Read ItMaster t Talk to a Tutor

Explanation / Answer

fx = 3x^2 - 27y
fy = -27x + 81y^2

Equating them to 0 :
3x^2 - 27y = 0
x^2 = 9y

-27x + 81y^2 = 0
x = 3y^2

So, we have :
x^2 = 9y
x = 3y^2

9y^4 = 9y
9y^4 - 9y = 0
y^4 - y = 0
y(y^3 - 1) = 0
So, y = 0 or y = 1

And using x = 3y^2, we have :
x = 0 or x = 3

So, criticals are
(0,0) and (3,1)

Now, we classify em :
fxx = 9x
fyy = 162y
fxy = -27

Now, D = fxx*fyy - fxy^2
D = 1458xy - 729

With (0,0), we have D = -729 , which is < 0 ----> Saddle
With (3,1), we get D > 0 and fxx = 9x = 9*3 = 27 > 0
Since both D and fxx > 0, (3,1) is a minimum

Now, with (0,0), we get f = 0
With (3,1), we get f = -27

So, answers :
NONE NONE NONE
NONE NONE NONE

minimum :
f(3,1) = -27
NONE NONE NONE

Saddle :
(0,0)
NONE NONE

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