Finding Stationary Points of f(x,y) = xye^-x-y

[chicago106]

Homework Statement

Locate and classify all the stationary points of f(x,y) = xye^-x-y)


have i started this right?

dz/dx = y(-1-y)e^-x-y

dz/dy = x(-x-1)e^-x-y

if so what is the next step?

 

[Jmf]

So we have:

$$f(x,y) = xy e^{-(x+y)}$$

(I hope that's correct, I wasn't sure how to read your e^-x-y)

I think the derivatives of this are:

$$\frac{\partial f}{\partial x} = y(1-x)e^{-(x+y)}$$
$$\frac{\partial f}{\partial y} = x(1-y)e^{-(x+y)}$$

Which is similar to what you have, except the sign in front of the '1' is different.

Once you have the derivatives, do you know how to find a stationary point?

 

[chicago106]

So we have:

$$f(x,y) = xy e^{-(x+y)}$$

(I hope that's correct, I wasn't sure how to read your e^-x-y)

I think the derivatives of this are:

$$\frac{\partial f}{\partial x} = y(1-x)e^{-(x+y)}$$
$$\frac{\partial f}{\partial y} = x(1-y)e^{-(x+y)}$$

Which is similar to what you have, except the sign in front of the '1' is different.

Once you have the derivatives, do you know how to find a stationary point?

That is how you write, I was being lazy. Thanks for that, I can crack on with finding the stationary points.