Can someone show me the steps to solving these...

f(x,y) = x ln y + y ln x

df/dx =

df/dy =

df/dx^2 =

df/dy^2 =

df/dydx =

df/dxdy =

I believe df/dx = ln y + y/x

and df/dy = x/y + lnx

how would I find the critical points from that?

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- Jun 12th 2008, 10:17 AMvodkaPartial Derivatives
Can someone show me the steps to solving these...

f(x,y) = x ln y + y ln x

df/dx =

df/dy =

df/dx^2 =

df/dy^2 =

df/dydx =

df/dxdy =

I believe df/dx = ln y + y/x

and df/dy = x/y + lnx

how would I find the critical points from that? - Jun 12th 2008, 10:40 AMcolby2152
- Jun 12th 2008, 11:10 AMTheEmptySet
Finding the critical points is not easy.

I don't think it can be solved analytically(I could be wrong(Wink))

I drew a graph to see what it looked like and here it is.

Attachment 6763

So we want to solve the system of equations

subbing the first into the 2nd we get

finally we get

If we sub this back into the first equation we get

Finally we can use newtons method on the last equation

Based on the graph from above I used

After applying newtons method three times I got the values(on a calculator)

Oddly enough when I repeated the process but eliminted the x's and used newtons method again I got the same value for y.

so the critical numbers are )

I hope this helps.

P.S if anyone knows another way I would love to see it

Thanks, The Empty Set. - Jun 14th 2008, 06:21 PMTheEmptySet
I have been thinking about this and just noticed that

and

Are inverses of each other. So they are symmetric over the line y=x

This means that any of their intersections would have to be on the line y=x.

So if I sub this into the first equation I get

So the exact value is