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I need help founding Extrema of a Function

spynet201

New member
Hey Guys I hope All is fine ? Please I need Helping getting the points critiques And if it's minimum or maximum

Function : f(x;y) = xlny-ylnx
[h=3]Thanks[/h][h=3][/h]
 

Subhotosh Khan

Super Moderator
Hey Guys I hope All is fine ? Please I need Helping getting the points critiques And if it's minimum or maximum

Function : f(x;y) = xlny-ylnx
Thanks
What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33
 

spynet201

New member
this is what i get but in the exercice rule i must get extrema min or max so someting

[FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]log[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]log[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT]
we'll do it the long way first and then use a shortcut

[FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]log[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Math]x[/FONT]

[FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]log[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT]

set these both equal to zero and solve

[FONT=MathJax_Main]log[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Math]x[/FONT]

[FONT=MathJax_Main]log[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]y[/FONT]

The only solution to this is [FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]e[/FONT]

The shortcut is knowing that in symmetric expressions like this extrema always occur when [FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]x[/FONT]

If we set [FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]x[/FONT] we can immediately see [FONT=MathJax_Main]log[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1[/FONT]

[FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]e[/FONT]

To characterize this critical point we first look at

[FONT=MathJax_Math]D[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]y[/FONT] at the critical point

[FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT]

[FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]2[/FONT]

[FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]x[/FONT]

[FONT=MathJax_Math]D[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]e[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]e[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Size3]([/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Size3])[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]e[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]e[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]e[/FONT][FONT=MathJax_Main]2[/FONT]

[FONT=MathJax_Math]D[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Main]0[/FONT] and thus this critical point is a saddle point.
 

stapel

Super Moderator
Staff member
this is what i get but in the exercice rule i must get extrema min or max so someting...

[FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]log[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]log[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT]
we'll do it the long way first and then use a shortcut....
Do you perhaps mean, "This is what somebody else gave me in reply to another of my posts of this question, but he didn't give me everything I need for my hand-in homework, so you guys finish it for me"...?

How about you at least go to the effort of telling us how your book defines "the points critiques". Thank you! ;)
 

ksdhart2

Member
I'm a bit late to the thread it seems, but this exact question was also posted right here on this forum. If the original poster is actually interested in doing work for themselves, the hints provided there will hopefully be of some use.
 
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