Jmf said:So we have:
[tex]
f(x,y) = xy e^{-(x+y)}
[/tex]
(I hope that's correct, I wasn't sure how to read your e^-x-y)
I think the derivatives of this are:
[tex]
\frac{\partial f}{\partial x} = y(1-x)e^{-(x+y)}
[/tex]
[tex]
\frac{\partial f}{\partial y} = x(1-y)e^{-(x+y)}
[/tex]
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?
A stationary point is a point on a curve or surface where the tangent line or tangent plane is horizontal, meaning that the gradient or partial derivatives are equal to zero.
To find stationary points of a function, we need to first find the partial derivatives with respect to each variable (x and y). Then, set these partial derivatives equal to zero and solve for the values of x and y. These values correspond to the coordinates of the stationary points.
Finding stationary points is important because they can provide valuable information about the behavior of a function. They can indicate the maximum and minimum values of a function, as well as the points where the function changes from increasing to decreasing or vice versa.
Yes, a function can have more than one stationary point. In fact, a function can have multiple stationary points at different locations on the curve or surface.
To determine if a stationary point is a maximum or minimum, we can use the second derivative test. This involves finding the second partial derivatives and plugging in the coordinates of the stationary point. If the second derivative is positive, then the point is a minimum. If the second derivative is negative, then the point is a maximum. If the second derivative is zero, then the test is inconclusive and further analysis may be needed.