Find the local maxima and minima for##f(x,y) = x^3-xy-x+xy^3-y^4##

[chwala]

Homework Statement

see attached.

Relevant Equations

$\nabla f = 0$

Ok i have,

$f_x= 3x^2-y-1+y^3$

$f_y = -x+3xy^2-4y^3$

$f_{xx} = 6x$

$f_{yy} = 6xy - 12y^2$

$f_{xy} = -1+3y^2$

looks like one needs software to solve this?

I can see the solutions from wolframalpha: local maxima to two decimal places as;

$(x,y) = (-0.67, 0.43)$

...but i am more interested in steps that lead to the given solution...

 

[fresh_42]

looks like one needs software to solve this?

Indeed. See
https://www.wolframalpha.com/input?i=3x^2+y^3-y=1+AND+4y^3=x(3y^2-1)

I can see the solutions from wolframalpha: local maxima to two decimal places as;

$(x,y) = (-0.67, 0.43)$

...but i am more interested in steps that lead to the given solution...

Look at the plot. This gives you an idea of how a numerical algorithm could work. Walk along the blue line until you cross the orange line and determine whether it is a local minimum, a local maximum, or an inflection point.

 

[anuttarasammyak]

I confirmed by myself with Wolfram.

https://www.wolframalpha.com/input?i=solve+3x^2-y-1+y^3=0,+-x+3xy^2-4y^3=0

https://www.wolframalpha.com/input?i=get+maximum+and+minimum+of++z=x^3-xy-x+xy^3-y^4

https://www.wolframalpha.com/input?i=get+saddle+point+of++z=x^3-xy-x+xy^3-y^4

 

[chwala]

Indeed. See
https://www.wolframalpha.com/input?i=3x^2+y^3-y=1+AND+4y^3=x(3y^2-1)
Look at the plot. This gives you an idea of how a numerical algorithm could work. Walk along the blue line until you cross the orange line and determine whether it is a local minimum, a local maximum, or an inflection point.

Thanks from the plot we have the point $(-7.540, -5.595)$ being an inflection point or can we say saddle point? then $(0.471, -0.396)$ being the local minimum... bringing me to the next question, do we have a global maximum and global minimum for this problem?

 

[fresh_42]

Thanks from the plot we have the point $(-7.540, -5.595)$ being an inflection point or can we say saddle point?

Yes.

then $(0.471, -0.396)$ being the local minimum... bringing me to the next question, do we have a global maximum and global minimum for this problem?

Look at the links in @anuttarasammyak 's post #3.

 

[anuttarasammyak]

I see the three crossing points are extrema or saddle points in

https://www.wolframalpha.com/input?i=Plot+3x^2-y-1+y^3=0;-x+3xy^2-4y^3=0;x+=+0;6xy-12y^2+=+0