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March 9th, 2014, 10:08 AM   #1
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maximum and minimum of a function

Dear Mentor & Guru,

Help needed solve equation that attached to determine whether they are relative maximum, relative minimum or saddle points. Million Thanks
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March 9th, 2014, 10:51 AM   #2
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Re: maximum and minimum of a function

I'll do the first one... and you can do the second on your own.

First take partial derivative with respect to x, and set it to equal to 0:
2x-1/(4y) = 0

x=1/(8y) ... (first equation)

Do the same thing with respect to y:
2 + x/(4y^2) = 0

Substitute x from the first equation:
2 + 1/(32y^3) = 0
y^3 = -64
y=-4, and x=-1/32... This is your critical point coordinate.

To see what kind of points there, look here:
http://en.wikipedia.org/wiki/Second_par ... ative_test
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March 9th, 2014, 03:06 PM   #3
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Re: maximum and minimum of a function

Dear Mentor & Guru,
Million thanks for your guide and help to solve the question 1. I done the question 2 but not sure correct or not help to correct it if wrong and I don't know and understand how to determine whether they are relative maximum, relative minimum or saddle points. Please Help...
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March 9th, 2014, 04:00 PM   #4
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Re: maximum and minimum of a function

Partial derivative with respect to x: 6x^2 + 14x
Set it equal to 0:
6x^2+14x=0
2x(3x + 7)=0

2x=0 --> x=0
3x+7 --> x=-7/3

Partial derivative with respect to y: -8y
Set it equal to 0:
-8y=0 --> y=0

Coordinates of critical points:
(0,0) and (-7/3,0)

Take the partial derivative with respect to x, and then take its partial derivative with respect to x again, and let it be A. --> A=12x+14
Take the partial derivative with respect to x, and then take its partial derivative with respect to y, and let it be B. --> B = 0
Take the partial derivative with respect to y, and then take its partial derivative with respect to y again, and let it be C. --> C = -8

Nature of stationary points (plug in coordinates inside A, B, and C):
if AC - B^2 > 0 and A<0 , then local maximum
if AC - B^2 > 0 and A>0, then local minimum
if AC - B^2 < 0, then saddle point
if AC - B^2 = 0, undetermined

Point (0,0):
AC - B^2 = (12(0)+14)(- - 0 < 0
This means it's a saddle point.

Point (-7/3,0)
AC - B^2 = (12(-7/3)+14)(- - 0 > 0, and A = 12(-7/3)+14 < 0
This means it's a local maximum.
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March 9th, 2014, 07:19 PM   #5
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Re: maximum and minimum of a function

Dear Guru & Mentor,

Very appreciate that provide the full detail solution, Million Thanks. I'm more understand now to solve this equation. Thanks again and apologies if caused a lot trouble for you all.
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